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    <ul> <li><p>As for LP solver recommendations, two of the best are Gurobi and CPLEX (google them). They are free for academic users, and are capable of solving large-scale problems. I believe they have all the capabilities that you need. You can get sensitivity information (to a perturbation) from the shadow prices (i.e. the Lagrange multipliers).</p></li> <li><p>But I'm more interested in your original problem. As I understand it, it looks like this:</p></li> </ul> <p>Let S = {1,2,...,N} where N is the total number of functions. y is a scalar. f_{i}:R^{1} -> R^{1}.</p> <pre><code>minimize sum{i in S} (x_{i} * f_{i}(0)) x_{i} s.t. (1) sum {i in S} x_{i} * f_{i}(1) = 1 (2) sum {i in S} x_{i} * f_{i}(y) &gt;= 0 for all y in (2,inf] </code></pre> <ul> <li><p>It just seems to me that you might want to try solve this problem as an convex NLP rather than an LP. Large-scale interior point NLP solvers like IPOPT should be able to handle these problems easily. I strongly recommended trying IPOPT <a href="http://www.coin-or.org/Ipopt" rel="nofollow noreferrer">http://www.coin-or.org/Ipopt</a></p></li> <li><p>From a numerical point of view: for convex problems, warm-starting is not necessary with interior point solvers; and you don't have to worry about the combinatorial cycling of active sets. What you've described as "warm-starting" is actually perturbing the solution -- that's more akin to sensitivity analysis. In optimization parlance, warm-starting usually means supplying a solver with an initial guess -- the solver will take that guess and end up at the same solution, which isn't really what you want. The only exception is if the active set changes with a different initial guess -- but for a convex problem with a unique optimum, this cannot happen. </p></li> </ul> <p>If you need any more information, I'd be pleased to supply it.</p> <p>EDIT:</p> <p>Sorry about the non-standard notation -- I wish I could type in LaTeX like on MathOverflow.net. (Incidentally, you might try posting this there -- I think the mathematicians there would be interested in this problem)</p> <p>Ah now I see about the "y > 2". It isn't really an optimization constraint so much as an interval defining a space (I've edited my description above). My mistake. I'm wondering if you could somehow transform/project the problem from an infinite to a finite one? I can't think of anything right now, but I'm just wondering if that's possible.</p> <p>So your approach is to discretize the problem for y in (2,inf]. I'm guessing you're choosing a very big number to represent inf and a fine discretization grid. Oooo tricky. I suppose discretization is probably your best bet. Maybe guys who do real analysis have ideas.</p> <p>I've seen something similar being done for problems involving Lyapunov functions where it was necessary to enforce a property in every point within a convex hull. But that space was finite. </p>
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    1. POWhich linear programming package should I use for high numbers of constraints and "warm starts"
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    1. POWhich linear programming package should I use for high numbers of constraints and "warm starts"
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    1. COThanks for all the recommendations and advice! The notation you used to transcribe my problem is a little condensed, but it looks possibly correct. An important point is that the constraint sum {i in S} x_{i} * f_{i}(y) >= 0 must hold for ALL y>2. That is, the function F(y)=sum_i x_i f_i(y) must be nonnegative on the whole interval [2,infinity). You could say y is a continuous label for an infinite number of constraints. I'll definitely look into the programs you suggest, though I may have some followup questions if it's not clear how to adapt them to my particular problem. Thanks again!
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    2. COAh, I see. That's a good idea about trying MathOverflow -- I might do that. It is definitely a tricky problem, but I'm sure someone has come across it before. Thanks again for your time and advice!
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