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\title{CAQM: Convexity Analysis of Quadratic Maps}
\date{}
\author{Anatoly Dymarsky, Elena Gryazina, Sergei Volodin, Boris Polyak}

\begin{document}
\maketitle

\section{Additional functions in {\rm library/}}

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\begin{tabular}{|p{35mm}|c|p{50mm}|c|p{30mm}|c|}
	\hline
	\bf Name & \bf Input & \bf Call & \bf Description & \bf Return value & \bf Exception\\\hline
	\bf Random map & {Dimensions n, m} & {\tt get\_random\_f(n, m, is\_complex)} & Generates random map $f$ & {\tt [A, b]} & None\\\hline
	\bf Value at x & The point $x\in X$ & {\tt quadratic\_map(A, b, x)} & Calculates $f(x)$ & $y=f(x)$ & None\\\hline
	\bf Product $c\cdot A$ & Normal vector $c$ & {\tt get\_Ac(A, c)} & Calculates $c\cdot A$ & $A_c=c\cdot A$ & None\\\hline
	\bf Get $H_c$ & $c,y\in\R^m$ & {\tt get\_H\_c(A, b, c, y)} & Calculates $H_c=\left(\begin{array}{cc}A_c & b_c\\b_c' &-(c,y) \end{array}\right)$ & $H_c$ & None\\\hline
	\bf Minimize $z(c)$ & $c,c_+$, step $\beta$ & {\tt minimize\_z\_c(A, b, c, c\_plus, beta\_initial, max\_step)} & Calculates $\inf\limits_{c\in C_-}z(c)$ & {\tt [z, c\_array, z\_array]} & If failed\\\hline
	\bf  $\R^n$ projection & & {\tt project(A, b, c, x\_0, delta\_c, normal, search\_area\_size)} & Projects $c+\Delta c$ to $C_-$ & {\tt [c\_new, lambda]} & If failed\\\hline
	\bf $\mathbb{C}^n$ projection & & {\tt project\_descent(A, b, c, normal\_1, normal\_2)} & Projects $c$ to $C_-$ & {\tt [c\_new, distance]} & If failed\\\hline
	\bf Gradient $\frac{\partial z}{\partial c}$ & Normal $c$ & {\tt get\_dz\_dc(A, b, c)} & Calculates $Q$, $\nabla z(c)$, normal vectors $n_1$, $n_2$ & {\tt [Q, Q\_inv, k, v, lambda\_min, z, dz\_dc, normal\_re, normal\_im, drho\_dc]} & None\\\hline
	\bf Change of basis & $c_+$ & {\tt change\_basis(A, b, c\_plus)} & Finds $\begin{cases}
	x = S(x'+x_0)\\
	y = y' + y_0
	\end{cases}
	$ s.t.
	$\begin{cases}
	c_+\cdot A_0=I\\
	c_+\cdot b_0=0
	\end{cases}$ & {\tt [A\_new, b\_new, x0, y0]} & None\\\hline
	\bf Boundary point $\partial F_c$ & Normal $c$ & {\tt boundary\_point(A, b, c)} & Calculates pre-image $x\colon f(x)\in\partial F_c$ & {\tt [x]} & If $c\cdot A<0$\\\hline
\end{tabular}

\newpage

\section{Numerical parameters in {\rm library/get\_config.m}}

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\begin{tabular}{|p{47mm}|p{27mm}|p{34mm}|p{130mm}|c|}
	\hline
	\bf Parameter name & \bf Target function & \bf Purpose & \bf Description & \bf Default value\\\hline
	{\tt Q\_inv\_eps} & {\tt get\_dz\_dc} & Pseudoinverse & Tolerance for {\tt pinv} function for $Q$ matrix & $10^{-5}$ \\\hline
	{\tt c\_plus\_min\_lambda} & {\tt is\_c\_plus} & $\lambda > 0$ check & Minimal eigenvalue which is still considered $> 0$ & $10^{-4}$ \\\hline
	{\tt c\_minus\_lambda\_min} & {\tt is\_nonconvex} & $\lambda > 0$ check & Maximal eigenvalue which is still considered to be $0$ & $10^{-7}$ \\\hline
	{\tt c\_minus\_h\_rank} & {\tt is\_nonconvex} & Rank check & Tolerance for {\tt rank} function in homogeneous case & $10^{-7}$ \\\hline
	{\tt c\_minus\_ortho} & {\tt is\_nonconvex} & Orthogonality criteria & Maximal value of scalar product of $|x_0^T (c \cdot b)|$ s.t. they are still considered orthogonal & $10^{-7}$ \\\hline
	{\tt c\_minus\_f1f2rank} & {\tt is\_nonconvex} & Rank check & Tolerance for {\tt rank} function for $f_1 \nparallel f_2$ check & $10^{-7}$ \\\hline
	{\tt descent\_min\_norm} & {\tt minimize\_z\_c} & Stopping criteria & Minimal norm of a vector (gradient, norm) & $10^{-3}$ \\\hline
	{\tt descent\_max\_z} & {\tt minimize\_z\_c} & Stopping criteria & Maximal value of $z$ & $10^{9}$ \\\hline
	{\tt descent\_rank\_Q} & {\tt minimize\_z\_c} & Stopping criteria & Tolerance for {\tt rank(Q)} check & $10^{-5}$ \\\hline
	{\tt descent\_min\_beta} & {\tt minimize\_z\_c} & Stopping criteria & Minimal step in direction of the gradient & $10^{-15}$ \\\hline
	{\tt descent\_theta} & {\tt minimize\_z\_c} & Step reduction & Multiplicative factor to reduce beta if projection failed & $0.5$ \\\hline
	{\tt descent\_max\_cos} & {\tt minimize\_z\_c} & Stopping criteria & Maximal cos between grad and n s.t. they are still considered to be not parallel & $1-10^{-4}$ \\\hline
	{\tt project\_descent\_max\_dist} & {\tt project\_descent} & Stopping criteria & Maximal tolerable distance to $C_-$ as value of constraint & $10^{-10}$ \\\hline
	{\tt project\_descent\_min\_alpha} & {\tt project\_descent} & Stopping criteria & Minimal step in the direction of the gradient & $10^{-20}$ \\\hline
	{\tt project\_descent\_theta} & {\tt project\_descent} & Step reduction & Multiplicative factor to reduce alpha & $0.4$ \\\hline
	{\tt project\_bisection\_min\_rl} & {\tt project} & Stopping criteria & Minimal value of $|r - l|$ & $10^{-8}$ \\\hline
	{\tt project\_bisection\_min\_value} & {\tt project} & Stopping criteria & Minimal value of distance to $C_-$ s.t. considered inside this set & $10^{-9}$ \\\hline
	{\tt project\_bisection\_max\_value} & {\tt project} & Precondition & Maximal value of the initial distance to $C_-$ s.t. projection is not attempted & $10^{-3}$ \\\hline
\end{tabular}

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