The feasibility of the waveform optimization problem is extensively studied , and a solution technique leading to an optimal waveform is proposed . The procedure requires the relaxation of the original problem into a convex optimization problem and involves a polynomial computational complexity . At the analysis stage , the waveform performance is studied in terms of trade-off among the achievable signal to interference plus noise ratio ( SINR ) , spectral shape , and the resulting autocorrelation function ( ACF ) . Algorithms for convex optimization with applications to data science Scott Roy Chair of the Supervisory Committee : Assistant Professor Dmitriy Drusvyatskiy Department of Mathematics Convex optimization is more popular than ever , with extensive applications in statistics , machine learning , and engineering . Nesterov introduced optimal first-order methods for large scale convex optimization in the 1980s , and extremely fast interior point methods for small-to-medium scale convex optimization emerged in the 1990s . The learning algorithm for these machines seeks the least informative distribution that will classify the training data correctly with high probability . One formulation can be expressed as a convex constrained optimization problem whose solution can be represented linearly in terms of training instances and their inner and outer products , supporting kernelization . The algorithm admits a natural PAC-Bayesian justification and is shown to minimize a quantity directly related to a PAC-Bayesian generalization bound . The control law design parameters are the tap weights of an FIR discretization of the Q-parameter . The multi-constraint H/sup 2//H/sup /spl infin// control law synthesis is written as a convex constrained optimization problem and solved as a structured semi-definite program . As a design example , we control the acoustic radiation from a ( mathematically modeled ) submerged spherical shell . However , very few research attempts have been directed toward studying the properties of SVDD solutions . In this work , the primal optimization of SVDD is first transformed into a convex constrained optimization problem , and the uniqueness of the centre of ball is proved while the non-uniqueness of the radius is investigated . This paper also investigates the property of the centre and radius from the perspective of the dual optimization problem , and suggests a method to calculate the radius . To overcome this problem , we propose a robust implementation of deterministic STAP in order to take into account a possible mismatch between the nominal and the actual target parameters . The proposed approach reformulates the deterministic STAP problem in the context of convex problem optimization . A detailed analysis of the maximum acceptable target parameters error is conducted , which ensures the existence of a numerical solution for the convex problem optimization . The proposed robust deterministic approach is defined for both the one dimensional ( spatial-only ) and the two dimensional ( space-time ) case . We aim to minimize the average file delivery time by investigating the optimal prefetching policy , i.e. , the size of the pre-fetched part of each file and the size of the dynamic storage . We formulate the file delivery time minimization problem as a convex optimization formulation . We compute the closed-form of the optimal pre-fetching policy when the wireless rate is constant . For the convenience of the filter design , a decoupling algorithm was applied to transform the obtained sufficient condition into a linear matrix inequality(LMI ) which can be easily solved . Furthermore , the filter design problem was changed to a convex optimization solving problem constrained by LMI to achieve good H∞ performances . A numerical example demonstrates the feasibility of the proposed filter design method . Compressive holography is a relatively time-consuming image estimation in convex optimized problem . We propose an efficient block-wise algorithm to limit the searching space and reduce the calculation time while keeping the reconstruction quality . The procedure takes advantage from the compressive sensing theory and is able to deal with linear , planar , and conformal fixed-geometry arrays . In the case of low faulty element percentages , the overall problem is reduced to a convex programming optimization . The minimum number of measurements allowing the success of recovery of either real or complex excitations is identified through an extensive set of numerical results . Shallow softmax regression models are also vulnerable to adversarial examples . Training on adversarial examples can regularize the model — however , this was not practical at the time due to the need for expensive constrained optimization in the inner loop . These results suggest that classifiers based on modern machine learning techniques , even those that obtain excellent performance on the test set , are not learning the true underlying concepts that determine the correct output label . The invention provides an ocean structure dynamic positioning thrust distributing method adopting dynamic prohibiting angles . The thrust distributing method is of a quadratic programming problem in an optimization problem ; the size of a prohibiting angle of a revolution angle of a front thruster , namely , a feasible region of the revolution angle , in the optimization problem is updated in real time by considering a speed ratio of front and rear full revolution thrusters ; and subsequently the quadratic programming problem is solved so as to obtain a thrust distribution scheme on the full revolution thruster . By introducing in the concept of the dynamic prohibiting angle , on one hand , the interference of the wake flow of the front thruster to the rear thruster is avoided , and on the other hand , the feasible region of the revolution angle of the front thruster is expanded , so that the quadratic programming problem can obtain more optimal and reasonable results . Perturbation subspaces for block eigenvector matrices are used to reduce the modified problem to a sequence of problems of smaller dimension . These perturbation subspaces are shown to be contained in certain generalized Krylov subspaces of the n-dimensional space , where n is the undoubled dimension of the matrices in the quadratic problem . The method converges at least as fast as the corresponding Taylor series , and the convergence can be accelerated further by applying a block generalization of the quadratically convergent Rayleigh quotient iteration . The linear problem is solved by an enhanced version of the Nashold projections . The quadratic problem is addressed by eigenvalue decompositions and quadratic programming methods . The general nonlinear formulation is solved by the local variations and gradient descent methods . In addition , it is shown that such a terminal set can be computed in a finite number of iterations . The state evolution over the prediction horizon is bounded by polytopic tubes and the MPC algorithm is formulated as a quadratic problem . Moreover , offset-free tracking can be achieved when the nominal plant matches the true plant . This paper investigates relaxations of parameterized LMI problems into standard LMI problems using techniques relying on directional convexity concepts . An in-depth discussion of the impact of the proposed techniques in quadratic programming , Lyapunov-based stability and performance analysis , $ \mu$ analysis , and linear parameter-varying control is provided . Illustrative examples are given to demonstrate the usefulness and practicality of the approach . Constraints of the problem for phase-lead and lag structures are derived . The objective function with the constraints is formed as a quadratic mathematical programming problem . For robust design , the parameters of the stabilizer are calculated under various operating conditions . The contact dynamic problem is formulated as force-based problem under the assumption of associative flow rule for displacement rates at contact points . The formulation of the variational problem associated to dynamics of the block assemblages gives rise to a quadratic mathematical programming problem . To evaluate the accuracy and computational efficiency of the implemented formulation , applications to numerical case studies from the literature are presented . In this paper , we present a statistical power estimation method where estimation time and accuracy can be balanced by assigning smaller errors to the nodes with higher power dissipation and higher errors to the nodes with lower power dissipation . To calculate the error rates for individual nodes , a quadratic programming based problem will be formulated which incorporates the distribution data of all individual node switching activities . Also , an iterative statistical power estimation system will be presented . By abstracting robots and obstacles into critical point sets respectively , the distance between the robot and obstacles can be described in a simpler way , then the obstacle avoidance strategy is established in form of inequality constraints by general class-K functions . Using minimal-velocity-norm ( MVN ) scheme , the control problem is formulated as a quadratic-programming case under multiple constraints . Then a deep recurrent neural network considering system models is established to solve the QP problem online . Based on model predictive control method a linear error model was described . Then the optimization problem of minimizing the objective function was regarded as a quadratic programming one under the control constrains . The experimental results show the efficiency in terms of the trajectory tracking control problem for mobile robot . In order to adapt to the current multi-level dispatching management system and to promote the operational efficiency of interconnected electricity networks , this paper proposes a decomposition collaborative model based on optimal power flow theory . The model is a quadratic programming question used to solve optimal power flow model . The information of interchanging between regions is communication price and boundary nodal bus phase angle . We do not introduce or consider any explicit logic , aim to discover ( fuzzy ) logic rules , or attempt to include prior knowledge in the form of logical constraints . amos2017optnet Introduces OptNet , a neural network layer that solve quadratic programs using an efficient differentiable solver . OptNet is trained to solve 4x4 Sudokus amongst other problems and beats the deep convolutional network baseline as described in park2016can . It has recently been shown that an extremely wide array of robust controller design problems may be reduced to the problem of finding a feasible point under a Biaffine Matrix Inequality ( BMI ) constraint . The BMI feasibility problem is the bilinear version of the Linear ( Affine ) Matrix Inequality ( LMI ) feasibility problem , and may also be viewed as a bilinear extension to the Semidefinite Programming ( SDP ) problem . The BMI problem may be approached as a biconvex global optimization problem of minimizing the maximum eigenvalue of a biaffine combination of symmetric matrices . constraint . In this paper , SSC is formulated as a semidefinite programming ( SDP ) problem . In order to solve the high computational complexity of SDP , we present a dual algorithm based on the Lagrange dual formalization . It is based on the two-channel structural PR FB proposed by Phoong et al ( 1995 ) . The basic principle is to represent the K-regularity condition as a set of linear equality constraints in the design variables so that the least square and minimax design problems can be solved , respectively , as a quadratic programming problem with linear equality constraints ( QPLC ) and a semidefinite programming ( SDP ) problem . We also demonstrate that it is always possible to realize such FBs with sum-of-powers-of-two ( SOPOT ) coefficients while preserving the regularity constraints using Bernstein polynomials . A semidefinite program ( SDP ) is an optimization problem over n × n symmetric matrices where a linear function of the entries is to be minimized subject to linear equality constraints , and the condition that the unknown matrix is positive semidefinite . Standard techniques for solving SDP 's require O(n3 ) operations per iteration . SemiDefinite Program ( SDP ) is one of principal problems in mathematical programming . Its application range is very wide and covers some problems arising from control theory ; for example , a stability condition for differential inclusions and discrete-time optimal control problems . In this paper , we propose a second-order corrector interior-point algorithm for semidefinite programming ( SDP ) . This algorithm is based on the wide neighborhood . Abstract In this paper we study the limiting behavior of the central path for semidefinite programming ( SDP ) . We show that the central path is an analytic function of the barrier parameter even at the limit point , provided that the semidefinite program has a strictly complementary solution . Interior-point methods in semidefinite programming ( SDP ) require the solution of a sequence of linear systems which are used to derive the search directions . Safeguards are typically required in order to handle rank-deficient Jacobians and free variables . We consider the problem of estimating an unknown parameter vector x in a linear model that may be subject to uncertainties , where the vector x is known to satisfy a weighted norm constraint . We first assume that the model is known exactly and seek the linear estimator that minimizes the worst-case mean-squared error ( MSE ) across all possible values of x. We show that for an arbitrary choice of weighting , the optimal minimax MSE estimator can be formulated as a solution to a semidefinite programming problem ( SDP ) , which can be solved very efficiently . We then develop a closed form expression for the minimax MSE estimator for a broad class of weighting matrices and show that it coincides with the shrunken estimator of Mayer and Willke , with a specific choice of shrinkage factor that explicitly takes the prior information into account . In the difference-of-convex-functions iteration method , we directly solve for the beamforming vectors via quadratic programming ( QP ) , avoiding the matrix rank constraints . In each iteration of the proposed algorithms , we only solve one simple convex semidefinite program ( SDP ) or QP . Our simulation results confirm that the proposed algorithms converge quickly after a few iterations . Quadratically constrained quadratic programming ( QCQP ) with double-sided constraints has plenty of applications in signal processing as have been addressed in recent years . QCQP problems are hard to solve , in general , and they are typically approached by solving a semidefinite programming ( SDP ) relaxation followed by a postprocessing procedure . Existing postprocessing schemes include Gaussian randomization to generate an approximate solution , rank reduction procedure ( the so-called purification ) , and some specific rank-one matrix decomposition techniques to yield a globally optimal solution . A key challenge for accurate PSSE is the inherent nonlinearity of SCADA measurements in the system states . Recent proposals for static PSSE tackle this issue by exploiting hidden convexity structure and solving a semidefinite programming ( SDP ) relaxation . In this work , an online PSSE algorithm based on SDP relaxation is proposed , which enjoys a similar convexity advantage , while capitalizing on past measurements as well for improved performance . In this paper , we present an alternative approach based on results from robust optimization to solve the stochastic linear-quadratic control ( SLQC ) problem . In the unconstrained case , the problem may be formulated as a semidefinite optimization problem ( SDP ) . We show that we can reduce this SDP to optimization of a convex function over a scalar variable followed by matrix multiplication in the current state , thus yielding an approach that is amenable to closed-loop control and analogous to the Riccati equation in our framework . The metric is trained with the goal that local semantic structure of data with same label is preserved while the biased distance of differently labeled data is maximized . The learning process also reduces to a convex optimization by formulating it as a semidefinite problem ( SDP ) . Numerous experiments demonstrate the superiority of our BML method over several current states of art approaches on publicly available dataset . The non-convex problem can be transformed into a two-stage problem . The inner stage is a semidefinite program ( SDP ) problem based on the semidefinite relaxation ( SDR ) and the outer stage is a one dimensional line search problem . It is also shown in the proof that the SDR is always tight and yields a rank-one solution . Random convex programs ( RCPs ) are convex optimization problems subject to a finite number $ N$ of random constraints . The optimal objective value $ J^*$ of an RCP is thus a random variable . Random convex programs ( RCPs ) are convex optimization problems subject to a finite number of constraints that are extracted at random according to some probability distribution . The optimal objective of an RCP , and its associated optimal solution ( when it exists ) , are random variables : RCP theory is mainly concerned with providing probabilistic assessments on the probability of objective and constraint violation in random convex programs . In the companion paper [ 5 ] we introduced a theory for random convex programs ( RCPs ) , deriving tight upper bounds for the probability of optimal objective and constraint violation . In the present paper , we extend the basic setup to the case of RCPs with a-posteriori violated constraints ( RCPVs ) : a paradigm that permits to improve the optimal objective value while maintaining the probability of objective violation under control . Methods based on statistical learning theory are applicable to non-convex problems , but they usually are conservative in achieving the desired probabilistic guarantees . In this paper , we derive a novel scenario approach for a wide class of random non-convex programs , with a sample complexity similar to that of uncertain convex programs and with probabilistic guarantees that hold not only for the optimal solution of the scenario program , but for all feasible solutions inside a set of a-priori chosen complexity . We also address measure-theoretic issues for uncertain convex and non-convex programs . Approaches based on statistical learning theory are applicable for a certain class of non-convex problems , but they usually are conservative in terms of performance and are computationally demanding . In this paper , we present a novel scenario approach for a wide class of random non-convex programs . We provide a sample complexity similar to the one for uncertain convex programs , but valid for all feasible solutions inside a set of a-priori chosen complexity . Finally , we apply these bounds to an example of robust truss topology design . The findings in this paper are a first step towards an extension of previous results on continuous random convex programs to the case of problems with mixed-integer decision variables that naturally occur in many real-world applications . In this work , we extend the NLTV framework by using some information divergences to build new sparsity measures for signal recovery . This leads to a general convex formulation of optimization problems involving information divergences . We address these problems by means of fast parallel proximal algorithms . In particular , we prepare access to the elegant framework of conic quadratic programming . These convex optimation problems are very well-structured , herewith resembling linear programs and , hence , permitting the use of interior point methods ( Nesterov and Nemirovskii , 1993 ) . We show , subject to some standard constraint qualifications , that the operations of addition and restriction are continuous . These results are applied to convex well-posed optimization problems , as well as to convergence of approximated solutions in infinite dimensional convex programming , to linear functionals exposing convex sets , and to metric projections . Also , for any given function in IX , we obtain results regarding the convergence of its inf-convolution with smoothing kernels . With the SCA method , the nonconvex part of the SRM problem can be locally linearized to its first-order Taylor expansion . Then , relying on solving a series of convexified optimization problems , an iterative precoding algorithm is developed . Numerical simulations are also provided to demonstrate the proposed algorithm . Notice that K ( p gen ) is a convex function of p gen ( x ) by definition , and both the equality and inequality constraints are affine functions of p gen ( x ) . Thus , problem ( 2 ) is a convex optimization problem . What 's more , since ( i ) dom K is open , and ( ii ) there exists a feasible solution p gen = p data to ( 3 ) , by the refined Slater 's condition [ reference ] , page 226 ) , we can further verify that strong duality holds for ( 3 ) . Proposition 3.1 . By the KKT conditions of the convex problem ( 3 ) , at the global optimum , the optimal generator distribution p * gen matches the true data distribution p data , and the optimal discriminator c * ( x ) has the following form : Then , is defined as the projection of the matrix on the observed entries , namely and otherwise . The unique solution of the convex optimization problem in Eq . [ reference ] is found by employing an Alternating Direction Method of Multipliers - based algorithm . In our setting , the number of points is the number of tasks and is typically low ; in contrast , the dimensionality is the number of shared parameters and can be in the millions . We therefore use a different approach based on convex optimization , since ( 3 ) is a convex quadratic problem with linear constraints . Before we tackle the general case , let 's consider the case of two tasks . Although this is only applicable when T = 2 , this enables efficient application of the Frank - Wolfe algorithm [ reference ] since the line search can be solved analytically . Hence , we use Frank - Wolfe to solve the constrained optimization problem , using ( 4 ) as a subroutine for the line search . We give all the update equations for the Frank - Wolfe solver in Algorithm 2 . The proposed framework applied to linear elliptic problems agrees with the classical theory . We present applications of the proposed framework to various interesting convex optimization problems such as nonlinear elliptic problems , nonsmooth problems , and nonsharp problems .
The feasibility of the waveform optimization problem is extensively studied , and a solution technique leading to an optimal waveform is proposed . The procedure requires the relaxation of the original problem into a convex optimization problem and involves a polynomial computational complexity . At the analysis stage , the waveform performance is studied in terms of trade-off among the achievable signal to interference plus noise ratio ( SINR ) , spectral shape , and the resulting autocorrelation function ( ACF ) .
Algorithms for convex optimization with applications to data science Scott Roy Chair of the Supervisory Committee : Assistant Professor Dmitriy Drusvyatskiy Department of Mathematics Convex optimization is more popular than ever , with extensive applications in statistics , machine learning , and engineering . Nesterov introduced optimal first-order methods for large scale convex optimization in the 1980s , and extremely fast interior point methods for small-to-medium scale convex optimization emerged in the 1990s .
The learning algorithm for these machines seeks the least informative distribution that will classify the training data correctly with high probability . One formulation can be expressed as a convex constrained optimization problem whose solution can be represented linearly in terms of training instances and their inner and outer products , supporting kernelization . The algorithm admits a natural PAC-Bayesian justification and is shown to minimize a quantity directly related to a PAC-Bayesian generalization bound .
The control law design parameters are the tap weights of an FIR discretization of the Q-parameter . The multi-constraint H/sup 2//H/sup /spl infin// control law synthesis is written as a convex constrained optimization problem and solved as a structured semi-definite program . As a design example , we control the acoustic radiation from a ( mathematically modeled ) submerged spherical shell .
However , very few research attempts have been directed toward studying the properties of SVDD solutions . In this work , the primal optimization of SVDD is first transformed into a convex constrained optimization problem , and the uniqueness of the centre of ball is proved while the non-uniqueness of the radius is investigated . This paper also investigates the property of the centre and radius from the perspective of the dual optimization problem , and suggests a method to calculate the radius .
To overcome this problem , we propose a robust implementation of deterministic STAP in order to take into account a possible mismatch between the nominal and the actual target parameters . The proposed approach reformulates the deterministic STAP problem in the context of convex problem optimization . A detailed analysis of the maximum acceptable target parameters error is conducted , which ensures the existence of a numerical solution for the convex problem optimization . The proposed robust deterministic approach is defined for both the one dimensional ( spatial-only ) and the two dimensional ( space-time ) case .
We aim to minimize the average file delivery time by investigating the optimal prefetching policy , i.e. , the size of the pre-fetched part of each file and the size of the dynamic storage . We formulate the file delivery time minimization problem as a convex optimization formulation . We compute the closed-form of the optimal pre-fetching policy when the wireless rate is constant .
For the convenience of the filter design , a decoupling algorithm was applied to transform the obtained sufficient condition into a linear matrix inequality(LMI ) which can be easily solved . Furthermore , the filter design problem was changed to a convex optimization solving problem constrained by LMI to achieve good H∞ performances . A numerical example demonstrates the feasibility of the proposed filter design method .
Compressive holography is a relatively time-consuming image estimation in convex optimized problem . We propose an efficient block-wise algorithm to limit the searching space and reduce the calculation time while keeping the reconstruction quality .
The procedure takes advantage from the compressive sensing theory and is able to deal with linear , planar , and conformal fixed-geometry arrays . In the case of low faulty element percentages , the overall problem is reduced to a convex programming optimization . The minimum number of measurements allowing the success of recovery of either real or complex excitations is identified through an extensive set of numerical results .
Shallow softmax regression models are also vulnerable to adversarial examples . Training on adversarial examples can regularize the model — however , this was not practical at the time due to the need for expensive constrained optimization in the inner loop . These results suggest that classifiers based on modern machine learning techniques , even those that obtain excellent performance on the test set , are not learning the true underlying concepts that determine the correct output label .
The invention provides an ocean structure dynamic positioning thrust distributing method adopting dynamic prohibiting angles . The thrust distributing method is of a quadratic programming problem in an optimization problem ; the size of a prohibiting angle of a revolution angle of a front thruster , namely , a feasible region of the revolution angle , in the optimization problem is updated in real time by considering a speed ratio of front and rear full revolution thrusters ; and subsequently the quadratic programming problem is solved so as to obtain a thrust distribution scheme on the full revolution thruster . By introducing in the concept of the dynamic prohibiting angle , on one hand , the interference of the wake flow of the front thruster to the rear thruster is avoided , and on the other hand , the feasible region of the revolution angle of the front thruster is expanded , so that the quadratic programming problem can obtain more optimal and reasonable results .
Perturbation subspaces for block eigenvector matrices are used to reduce the modified problem to a sequence of problems of smaller dimension . These perturbation subspaces are shown to be contained in certain generalized Krylov subspaces of the n-dimensional space , where n is the undoubled dimension of the matrices in the quadratic problem . The method converges at least as fast as the corresponding Taylor series , and the convergence can be accelerated further by applying a block generalization of the quadratically convergent Rayleigh quotient iteration .
The linear problem is solved by an enhanced version of the Nashold projections . The quadratic problem is addressed by eigenvalue decompositions and quadratic programming methods . The general nonlinear formulation is solved by the local variations and gradient descent methods .
In addition , it is shown that such a terminal set can be computed in a finite number of iterations . The state evolution over the prediction horizon is bounded by polytopic tubes and the MPC algorithm is formulated as a quadratic problem . Moreover , offset-free tracking can be achieved when the nominal plant matches the true plant .
This paper investigates relaxations of parameterized LMI problems into standard LMI problems using techniques relying on directional convexity concepts . An in-depth discussion of the impact of the proposed techniques in quadratic programming , Lyapunov-based stability and performance analysis , $ \mu$ analysis , and linear parameter-varying control is provided . Illustrative examples are given to demonstrate the usefulness and practicality of the approach .
Constraints of the problem for phase-lead and lag structures are derived . The objective function with the constraints is formed as a quadratic mathematical programming problem . For robust design , the parameters of the stabilizer are calculated under various operating conditions .
The contact dynamic problem is formulated as force-based problem under the assumption of associative flow rule for displacement rates at contact points . The formulation of the variational problem associated to dynamics of the block assemblages gives rise to a quadratic mathematical programming problem . To evaluate the accuracy and computational efficiency of the implemented formulation , applications to numerical case studies from the literature are presented .
In this paper , we present a statistical power estimation method where estimation time and accuracy can be balanced by assigning smaller errors to the nodes with higher power dissipation and higher errors to the nodes with lower power dissipation . To calculate the error rates for individual nodes , a quadratic programming based problem will be formulated which incorporates the distribution data of all individual node switching activities . Also , an iterative statistical power estimation system will be presented .
By abstracting robots and obstacles into critical point sets respectively , the distance between the robot and obstacles can be described in a simpler way , then the obstacle avoidance strategy is established in form of inequality constraints by general class-K functions . Using minimal-velocity-norm ( MVN ) scheme , the control problem is formulated as a quadratic-programming case under multiple constraints . Then a deep recurrent neural network considering system models is established to solve the QP problem online .
Based on model predictive control method a linear error model was described . Then the optimization problem of minimizing the objective function was regarded as a quadratic programming one under the control constrains . The experimental results show the efficiency in terms of the trajectory tracking control problem for mobile robot .
In order to adapt to the current multi-level dispatching management system and to promote the operational efficiency of interconnected electricity networks , this paper proposes a decomposition collaborative model based on optimal power flow theory . The model is a quadratic programming question used to solve optimal power flow model . The information of interchanging between regions is communication price and boundary nodal bus phase angle .
We do not introduce or consider any explicit logic , aim to discover ( fuzzy ) logic rules , or attempt to include prior knowledge in the form of logical constraints . amos2017optnet Introduces OptNet , a neural network layer that solve quadratic programs using an efficient differentiable solver . OptNet is trained to solve 4x4 Sudokus amongst other problems and beats the deep convolutional network baseline as described in park2016can .
It has recently been shown that an extremely wide array of robust controller design problems may be reduced to the problem of finding a feasible point under a Biaffine Matrix Inequality ( BMI ) constraint . The BMI feasibility problem is the bilinear version of the Linear ( Affine ) Matrix Inequality ( LMI ) feasibility problem , and may also be viewed as a bilinear extension to the Semidefinite Programming ( SDP ) problem . The BMI problem may be approached as a biconvex global optimization problem of minimizing the maximum eigenvalue of a biaffine combination of symmetric matrices .
constraint . In this paper , SSC is formulated as a semidefinite programming ( SDP ) problem . In order to solve the high computational complexity of SDP , we present a dual algorithm based on the Lagrange dual formalization .
It is based on the two-channel structural PR FB proposed by Phoong et al ( 1995 ) . The basic principle is to represent the K-regularity condition as a set of linear equality constraints in the design variables so that the least square and minimax design problems can be solved , respectively , as a quadratic programming problem with linear equality constraints ( QPLC ) and a semidefinite programming ( SDP ) problem . We also demonstrate that it is always possible to realize such FBs with sum-of-powers-of-two ( SOPOT ) coefficients while preserving the regularity constraints using Bernstein polynomials .
A semidefinite program ( SDP ) is an optimization problem over n × n symmetric matrices where a linear function of the entries is to be minimized subject to linear equality constraints , and the condition that the unknown matrix is positive semidefinite . Standard techniques for solving SDP 's require O(n3 ) operations per iteration .
SemiDefinite Program ( SDP ) is one of principal problems in mathematical programming . Its application range is very wide and covers some problems arising from control theory ; for example , a stability condition for differential inclusions and discrete-time optimal control problems .
In this paper , we propose a second-order corrector interior-point algorithm for semidefinite programming ( SDP ) . This algorithm is based on the wide neighborhood .
Abstract In this paper we study the limiting behavior of the central path for semidefinite programming ( SDP ) . We show that the central path is an analytic function of the barrier parameter even at the limit point , provided that the semidefinite program has a strictly complementary solution .
Interior-point methods in semidefinite programming ( SDP ) require the solution of a sequence of linear systems which are used to derive the search directions . Safeguards are typically required in order to handle rank-deficient Jacobians and free variables .
We consider the problem of estimating an unknown parameter vector x in a linear model that may be subject to uncertainties , where the vector x is known to satisfy a weighted norm constraint . We first assume that the model is known exactly and seek the linear estimator that minimizes the worst-case mean-squared error ( MSE ) across all possible values of x. We show that for an arbitrary choice of weighting , the optimal minimax MSE estimator can be formulated as a solution to a semidefinite programming problem ( SDP ) , which can be solved very efficiently . We then develop a closed form expression for the minimax MSE estimator for a broad class of weighting matrices and show that it coincides with the shrunken estimator of Mayer and Willke , with a specific choice of shrinkage factor that explicitly takes the prior information into account .
In the difference-of-convex-functions iteration method , we directly solve for the beamforming vectors via quadratic programming ( QP ) , avoiding the matrix rank constraints . In each iteration of the proposed algorithms , we only solve one simple convex semidefinite program ( SDP ) or QP . Our simulation results confirm that the proposed algorithms converge quickly after a few iterations .
Quadratically constrained quadratic programming ( QCQP ) with double-sided constraints has plenty of applications in signal processing as have been addressed in recent years . QCQP problems are hard to solve , in general , and they are typically approached by solving a semidefinite programming ( SDP ) relaxation followed by a postprocessing procedure . Existing postprocessing schemes include Gaussian randomization to generate an approximate solution , rank reduction procedure ( the so-called purification ) , and some specific rank-one matrix decomposition techniques to yield a globally optimal solution .
A key challenge for accurate PSSE is the inherent nonlinearity of SCADA measurements in the system states . Recent proposals for static PSSE tackle this issue by exploiting hidden convexity structure and solving a semidefinite programming ( SDP ) relaxation . In this work , an online PSSE algorithm based on SDP relaxation is proposed , which enjoys a similar convexity advantage , while capitalizing on past measurements as well for improved performance .
In this paper , we present an alternative approach based on results from robust optimization to solve the stochastic linear-quadratic control ( SLQC ) problem . In the unconstrained case , the problem may be formulated as a semidefinite optimization problem ( SDP ) . We show that we can reduce this SDP to optimization of a convex function over a scalar variable followed by matrix multiplication in the current state , thus yielding an approach that is amenable to closed-loop control and analogous to the Riccati equation in our framework .
The metric is trained with the goal that local semantic structure of data with same label is preserved while the biased distance of differently labeled data is maximized . The learning process also reduces to a convex optimization by formulating it as a semidefinite problem ( SDP ) . Numerous experiments demonstrate the superiority of our BML method over several current states of art approaches on publicly available dataset .
The non-convex problem can be transformed into a two-stage problem . The inner stage is a semidefinite program ( SDP ) problem based on the semidefinite relaxation ( SDR ) and the outer stage is a one dimensional line search problem . It is also shown in the proof that the SDR is always tight and yields a rank-one solution .
Random convex programs ( RCPs ) are convex optimization problems subject to a finite number $ N$ of random constraints . The optimal objective value $ J^*$ of an RCP is thus a random variable .
Random convex programs ( RCPs ) are convex optimization problems subject to a finite number of constraints that are extracted at random according to some probability distribution . The optimal objective of an RCP , and its associated optimal solution ( when it exists ) , are random variables : RCP theory is mainly concerned with providing probabilistic assessments on the probability of objective and constraint violation in random convex programs .
In the companion paper [ 5 ] we introduced a theory for random convex programs ( RCPs ) , deriving tight upper bounds for the probability of optimal objective and constraint violation . In the present paper , we extend the basic setup to the case of RCPs with a-posteriori violated constraints ( RCPVs ) : a paradigm that permits to improve the optimal objective value while maintaining the probability of objective violation under control .
Methods based on statistical learning theory are applicable to non-convex problems , but they usually are conservative in achieving the desired probabilistic guarantees . In this paper , we derive a novel scenario approach for a wide class of random non-convex programs , with a sample complexity similar to that of uncertain convex programs and with probabilistic guarantees that hold not only for the optimal solution of the scenario program , but for all feasible solutions inside a set of a-priori chosen complexity . We also address measure-theoretic issues for uncertain convex and non-convex programs .
Approaches based on statistical learning theory are applicable for a certain class of non-convex problems , but they usually are conservative in terms of performance and are computationally demanding . In this paper , we present a novel scenario approach for a wide class of random non-convex programs . We provide a sample complexity similar to the one for uncertain convex programs , but valid for all feasible solutions inside a set of a-priori chosen complexity .
Finally , we apply these bounds to an example of robust truss topology design . The findings in this paper are a first step towards an extension of previous results on continuous random convex programs to the case of problems with mixed-integer decision variables that naturally occur in many real-world applications .
In this work , we extend the NLTV framework by using some information divergences to build new sparsity measures for signal recovery . This leads to a general convex formulation of optimization problems involving information divergences . We address these problems by means of fast parallel proximal algorithms .
In particular , we prepare access to the elegant framework of conic quadratic programming . These convex optimation problems are very well-structured , herewith resembling linear programs and , hence , permitting the use of interior point methods ( Nesterov and Nemirovskii , 1993 ) .
We show , subject to some standard constraint qualifications , that the operations of addition and restriction are continuous . These results are applied to convex well-posed optimization problems , as well as to convergence of approximated solutions in infinite dimensional convex programming , to linear functionals exposing convex sets , and to metric projections . Also , for any given function in IX , we obtain results regarding the convergence of its inf-convolution with smoothing kernels .
With the SCA method , the nonconvex part of the SRM problem can be locally linearized to its first-order Taylor expansion . Then , relying on solving a series of convexified optimization problems , an iterative precoding algorithm is developed . Numerical simulations are also provided to demonstrate the proposed algorithm .
Notice that K ( p gen ) is a convex function of p gen ( x ) by definition , and both the equality and inequality constraints are affine functions of p gen ( x ) . Thus , problem ( 2 ) is a convex optimization problem . What 's more , since ( i ) dom K is open , and ( ii ) there exists a feasible solution p gen = p data to ( 3 ) , by the refined Slater 's condition [ reference ] , page 226 ) , we can further verify that strong duality holds for ( 3 ) .
Proposition 3.1 . By the KKT conditions of the convex problem ( 3 ) , at the global optimum , the optimal generator distribution p * gen matches the true data distribution p data , and the optimal discriminator c * ( x ) has the following form :
Then , is defined as the projection of the matrix on the observed entries , namely and otherwise . The unique solution of the convex optimization problem in Eq . [ reference ] is found by employing an Alternating Direction Method of Multipliers - based algorithm .
In our setting , the number of points is the number of tasks and is typically low ; in contrast , the dimensionality is the number of shared parameters and can be in the millions . We therefore use a different approach based on convex optimization , since ( 3 ) is a convex quadratic problem with linear constraints . Before we tackle the general case , let 's consider the case of two tasks .
Although this is only applicable when T = 2 , this enables efficient application of the Frank - Wolfe algorithm [ reference ] since the line search can be solved analytically . Hence , we use Frank - Wolfe to solve the constrained optimization problem , using ( 4 ) as a subroutine for the line search . We give all the update equations for the Frank - Wolfe solver in Algorithm 2 .
The proposed framework applied to linear elliptic problems agrees with the classical theory . We present applications of the proposed framework to various interesting convex optimization problems such as nonlinear elliptic problems , nonsmooth problems , and nonsharp problems .
Radar waveform design in a spectrally crowded environment via nonconvex quadratic optimization Algorithms for convex optimization with applications to data science Gaussian Margin Machines A practical method for constrained-optimization controller design: H2 or H/spl infin/ optimization with multiple H2 and/or H/spl infin/ constraints Theoretical Analysis for the Optimization Problem of Support Vector Data Description A Robust Direct Data Domain Approach for STAP Fractional Dynamic Caching: A Collaborative Design of Storage and Backhaul Decentralized H_∞ Filtering for Interconnected Time-Delay Systems Efficient block-wise algorithm for compressive holography. A Compressive-Sensing-Inspired Procedure for Array Antenna Diagnostics by a Small Number of Phaseless Measurements Explaining and Harnessing Adversarial Examples Dynamic positioning thrust distributing device adopting dynamic prohibiting angle and distributing method thereof A Subspace Approximation Method for the Quadratic Eigenvalue Problem Fast pixel-based mask optimization for inverse lithography Robust Tube-Based Tracking MPC for Linear Systems with Multiplicative Uncertainty Parameterized LMIs in Control Theory Robust design of static synchronous series compensator-based stabilizer for damping inter-area oscillations using quadratic mathematical programming INVITED: A simple contact model for dynamic analysis of masonry block structures using mathematical programming An optimization-based error calculation for statistical power estimation of CMOS logic circuits Deep Recurrent Neural Networks Based Obstacle Avoidance Control for Redundant Manipulators Trajectory Tracking for Mobile Robot Using Model Predictive Control Decomposition Model Based on Optimal Power Flow in Interconnected Electricity Networks Recurrent Relational Networks Global optimization for the Biaffine Matrix Inequality problem Efficient Semidefinite Spectral Clustering via Lagrange Duality New design and realization techniques for a class of perfect reconstruction two-channel FIR filterbanks and wavelets bases A Subspace Semidefinite Programming for Spectral Graph Partitioning Efficient parallel software for large-scale Semidefinite Programs A new second-order corrector interior-point algorithm for semidefinite programming Analyticity of the central path at the boundary point in semidefinite programming A primal–dual regularized interior-point method for semidefinite programming Robust mean-squared error estimation in the presence of model uncertainties Secure beamforming for max-min SINR in multi-cell SWIPT systems Randomized Algorithms for Optimal Solutions of Double-Sided QCQP With Applications in Signal Processing Online semidefinite programming for power system state estimation Constrained Stochastic LQC: A Tractable Approach Biased metric learning for person-independent head pose estimation Secure beamforming design in wiretap MISO interference channels Random Convex Programs Random convex programs: Dealing with the non-existing solution nuisance Random convex programs part 2 A Scenario Approach for Non-Convex Control Design A scenario approach to non-convex control design: Preliminary probabilistic guarantees On mixed-integer random convex programs A proximal approach for signal recovery based on information measures Organization in Finance Prepared by Stochastic Differential Equations with Additive and Nonlinear Models and Continuous Optimization The EPI-Distance Topology: Continuity and Stability Results with Applications to Convex Optimization Problems AN-Aided Secrecy Precoding for SWIPT in Cognitive MIMO Broadcast Channels Calibrating Energy-based Generative Adversarial Networks 3D Face Morphable Models "In-the-Wild" Multi-Task Learning as Multi-Objective Optimization Additive Schwarz Methods for Convex Optimization as Gradient Methods
af9a6ec5b5bffea83d13e49c71f8ffe284545a52 47af82d2ee4ecb3ee5aa549984d1cc8939f88b9f 5633dfd7d9989e53e83c0d22f77b09581caea43d f118bd959bffe343ce19a621167aa455c9058bb3 4ca365df8d4ffe5f3b833160d3385bba9a5673ce f7cf7319599c11ecb95697013ed417f0b7cb897d c5c72b42863311e45f6ffe181d090f87bf83f72b 5eabae6249881b324f1a2d5a57d16dea0ae03673 df5c8c575aca14966582572393a2cc4dff8344e5 0a260390f90cced7ad96bc8b21f4b3ca67956394 34f1bfb780d2d66af8b3793a681935a01928cc56 069fb37e9bdbff7c0727b531dc65bf2b2a889e9d afa04bdc0c4ffbba81cfe26ebe3da177710e8d36 69ead31d30a2ff85defb8cece84845b097af2a4a ffc6046b8b396c5eeef178725a9dde02db107d47 a9ee4296a572130397a54cb4cc2018d49cfd525e 82ab75ac9c9cd9d40a667179a22ffd3e1ee50b72 c402a6252c671203cf17203d7b0f4dd254f808bf 7cde073203153907274d2b895b85f0122f566623 024e8124793684ac0c404199f611540a491aa002 6beff715f36b5ab2bd9016f1dbcc95ecee652d32 7eb320c323e205e02d40ca29472a944d57c1874c e2469ad13161fb3809e6823ee397e40f1b261953 b6badfe989aa1c0bb018de7fd432e58760c969b4 bd217aed581532cb996df04be91e0e14ddfbe9d4 a7fe20117a0819311c83d1bfe4983b43bc5ae4b7 2594e47cfa18e6cf7867d72992fb90d2b3b7bd71 c888cb8c713f89d6e6d972bef74c3f3496ce86c6 946f60103f07244fa513b29e4e1ed75aef50d9d2 3109e6aa23738668050f1494d1c381e4a7d42646 afda8b924c795fc2a4836b14fc89721997aec0d1 acbaeab0a2ea0bbe91ca18835fe34db36d66b503 b9410290a42f3871951a9dcdd41213e1a1cdc4d1 9fd1d43087ca88953dbba4b4886f1225302d736a 38e627f69bfc960ab00117eff20e1b7994c3f601 b35bfae0815d3654c380966d495e61d4ab611cd3 93d6ee1abbaba9b8788475d5edda04fc878b3648 64902f4cb6424c82eceb22783c8551e56a475cfe f3b1f0b3c8a4fa00a895556dba0e0373a93d8c2e 7f9938d81cb22e3498aa4ce2585e2aafebb794e4 d53124abe5bda3230d267e52be2fbff246b1b7e1 6a2248f66feacb1bfb9de5fa4f3dcd13e018cdf7 c6d61f6e9167e0caa25c5b6f9facf3a1449d6764 dd51ed5c240ae2a74039e004d037fb594f3961c7 a3f9d3c7bcefff2cf74a4638b7d3eb4a82e395b7 b63b2a71b12f8b608a4835a88e02f7d481ed8432 aeb83e5c88da42c965e2092abbc488ceb98b4404
["Mathematics","Computer Science"] ["Computer Science"] ["Computer Science"] ["Computer Science"] ["Computer Science"] ["Computer Science","Mathematics"] ["Computer Science"] ["Computer Science"] ["Medicine","Physics"] ["Physics"] ["Computer Science"] ["Computer Science","Mathematics"] ["Physics"] ["Computer Science"] ["Computer Science","Mathematics"] ["Computer Science"] ["Computer Science"] ["Computer Science"] ["Medicine","Computer Science"] ["Computer Science"] ["Engineering"] ["Mathematics","Computer Science"] ["Mathematics","Computer Science","Medicine"] ["Mathematics","Computer Science"] ["Computer Science"] ["Mathematics","Computer Science"] ["Computer Science","Mathematics"] ["Computer Science","Mathematics"] ["Computer Science","Mathematics"] ["Mathematics","Computer Science"] ["Computer Science"] ["Mathematics","Computer Science"] ["Computer Science"] ["Computer Science","Mathematics"] ["Computer Science"] ["Computer Science"] ["Mathematics","Computer Science"] ["Computer Science"] ["Computer Science","Mathematics"] ["Computer Science","Mathematics"] ["Computer Science"] ["Mathematics","Computer Science"] ["Mathematics","Computer Science"] ["Economics"] ["Mathematics","Computer Science"] ["Computer Science"] ["Mathematics","Computer Science"]
2014 2017 2009 1995 2011 2012 2020 2011 2017 2016 2015 2012 2005 2006 2019 2000 2010 2018 1998 2019 2009 2010 2018 1995 2014 2004 2002 2010 2012 2002 2012 2005 2016 2014 2014 2007 2011 2015 2010 2010 2010 2016 2014 2012 2013 2008 1992 2015 2017 2017 2018 2020
AISTATS ICLR CDC DAC NeurIPS International Conference on Computational Science (1) CACSD WCNC Workshops ICASSP ACM Multimedia ICCC CDC CACSD ACC EUSIPCO ICLR 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) NeurIPS
https://api.semanticscholar.org/af9a6ec5b5bffea83d13e49c71f8ffe284545a52 https://api.semanticscholar.org/47af82d2ee4ecb3ee5aa549984d1cc8939f88b9f https://api.semanticscholar.org/5633dfd7d9989e53e83c0d22f77b09581caea43d https://api.semanticscholar.org/f118bd959bffe343ce19a621167aa455c9058bb3 https://api.semanticscholar.org/4ca365df8d4ffe5f3b833160d3385bba9a5673ce https://api.semanticscholar.org/f7cf7319599c11ecb95697013ed417f0b7cb897d https://api.semanticscholar.org/c5c72b42863311e45f6ffe181d090f87bf83f72b https://api.semanticscholar.org/5eabae6249881b324f1a2d5a57d16dea0ae03673 https://api.semanticscholar.org/df5c8c575aca14966582572393a2cc4dff8344e5 https://api.semanticscholar.org/0a260390f90cced7ad96bc8b21f4b3ca67956394 https://www.semanticscholar.org/paper/bee044c8e8903fb67523c1f8c105ab4718600cdb https://api.semanticscholar.org/34f1bfb780d2d66af8b3793a681935a01928cc56 https://api.semanticscholar.org/069fb37e9bdbff7c0727b531dc65bf2b2a889e9d https://api.semanticscholar.org/afa04bdc0c4ffbba81cfe26ebe3da177710e8d36 https://api.semanticscholar.org/69ead31d30a2ff85defb8cece84845b097af2a4a https://api.semanticscholar.org/ffc6046b8b396c5eeef178725a9dde02db107d47 https://api.semanticscholar.org/a9ee4296a572130397a54cb4cc2018d49cfd525e https://api.semanticscholar.org/82ab75ac9c9cd9d40a667179a22ffd3e1ee50b72 https://api.semanticscholar.org/c402a6252c671203cf17203d7b0f4dd254f808bf https://api.semanticscholar.org/7cde073203153907274d2b895b85f0122f566623 https://api.semanticscholar.org/024e8124793684ac0c404199f611540a491aa002 https://api.semanticscholar.org/6beff715f36b5ab2bd9016f1dbcc95ecee652d32 https://www.semanticscholar.org/paper/34273979fd2a62fd7b49ee6d14a925864ff94e74 https://api.semanticscholar.org/7eb320c323e205e02d40ca29472a944d57c1874c https://api.semanticscholar.org/e2469ad13161fb3809e6823ee397e40f1b261953 https://api.semanticscholar.org/b6badfe989aa1c0bb018de7fd432e58760c969b4 https://api.semanticscholar.org/bd217aed581532cb996df04be91e0e14ddfbe9d4 https://api.semanticscholar.org/a7fe20117a0819311c83d1bfe4983b43bc5ae4b7 https://api.semanticscholar.org/2594e47cfa18e6cf7867d72992fb90d2b3b7bd71 https://api.semanticscholar.org/c888cb8c713f89d6e6d972bef74c3f3496ce86c6 https://api.semanticscholar.org/946f60103f07244fa513b29e4e1ed75aef50d9d2 https://api.semanticscholar.org/3109e6aa23738668050f1494d1c381e4a7d42646 https://api.semanticscholar.org/afda8b924c795fc2a4836b14fc89721997aec0d1 https://api.semanticscholar.org/acbaeab0a2ea0bbe91ca18835fe34db36d66b503 https://api.semanticscholar.org/b9410290a42f3871951a9dcdd41213e1a1cdc4d1 https://api.semanticscholar.org/9fd1d43087ca88953dbba4b4886f1225302d736a https://api.semanticscholar.org/38e627f69bfc960ab00117eff20e1b7994c3f601 https://api.semanticscholar.org/b35bfae0815d3654c380966d495e61d4ab611cd3 https://api.semanticscholar.org/93d6ee1abbaba9b8788475d5edda04fc878b3648 https://api.semanticscholar.org/64902f4cb6424c82eceb22783c8551e56a475cfe https://api.semanticscholar.org/f3b1f0b3c8a4fa00a895556dba0e0373a93d8c2e https://api.semanticscholar.org/7f9938d81cb22e3498aa4ce2585e2aafebb794e4 https://api.semanticscholar.org/d53124abe5bda3230d267e52be2fbff246b1b7e1 https://api.semanticscholar.org/6a2248f66feacb1bfb9de5fa4f3dcd13e018cdf7 https://api.semanticscholar.org/c6d61f6e9167e0caa25c5b6f9facf3a1449d6764 https://api.semanticscholar.org/dd51ed5c240ae2a74039e004d037fb594f3961c7 https://api.semanticscholar.org/a3f9d3c7bcefff2cf74a4638b7d3eb4a82e395b7 https://api.semanticscholar.org/b63b2a71b12f8b608a4835a88e02f7d481ed8432 https://www.semanticscholar.org/paper/0496691c6d54fd746c111d1004349acf7654145e https://www.semanticscholar.org/paper/4b2443858fa92c374c1fb0f3ac3c59b2886adf9f https://www.semanticscholar.org/paper/2b0d7e51efd004fe3847f54863540c79312f3546 https://api.semanticscholar.org/aeb83e5c88da42c965e2092abbc488ceb98b4404