title: Which linear programming package should I use for high numbers of constraints and "warm starts"
body: <p>I have a "continuous" linear programming problem that involves maximizing a linear function over a curved convex space.  In typical LP problems, the convex space is a polytope, but in this case the convex space is piecewise curved -- that is, it has faces, edges, and vertices, but the edges aren't straight and the faces aren't flat.  Instead of being specified by a finite number of linear inequalities, I have a continuously infinite number.  I'm currently dealing with this by approximating the surface by a polytope, which means discretizing the continuously infinite constraints into a very large finite number of constraints.</p>
  
  <p>I'm also in the situation where I'd like to know how the answer changes under small perturbations to the underlying problem.  Thus, I'd like to be able to supply an initial condition to the solver based on a nearby solution.  I believe this capability is called a "warm start."</p>
  
  <p>Can someone help me distinguish between the various LP packages out there?  I'm not so concerned with user-friendliness as speed (for large numbers of constraints), high-precision arithmetic, and warm starts.</p>
  
  <p>Thanks!</p>
  
  <p>EDIT: Judging from the conversation with question answerers so far, I should be clearer about the problem I'm trying to solve.  A simplified version is the following:</p>
  
  <p>I have N fixed functions f_i(y) of a single real variable y. I want to find x_i (i=1,...,N) that minimize \sum_{i=1}^N x_i f_i(0), subject to the constraints:</p>
  
  <ul>
  <li>\sum_{i=1}^N x_i f_i(1) = 1, and</li>
  <li>\sum_{i=1}^N x_i f_i(y) >= 0 for all y>2</li>
  </ul>
  
  <p>More succinctly, if we define the function F(y)=\sum_{i=1}^N x_i f_i(y), then I want to minimize F(0) subject to the condition that F(1)=1, and F(y) is positive on the entire interval [2,infinity).  Note that this latter positivity condition is really an <strong>infinite</strong> number of linear constraints on the x_i's, one for each y.  You can think of y as a label -- it is not an optimization variable.  A specific y_0 restricts me to the half-space F(y_0) >= 0 in the space of x_i's.  As I vary y_0 between 2 and infinity, these half-spaces change continuously, carving out a curved convex shape.  The geometry of this shape depends implicitly (and in a complicated way) on the functions f_i.</p>