# Questions tagged [optimality-conditions]

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What is the difference between Optimality cuts and Feasibility cuts for L shaped method in stochastic programming? Like for the following problem they used Optimality cuts, \begin{aligned} & z=\... 3 votes 0 answers 29 views ### How to establish the optimal value functions and optimal control policy for a controlled random walk problem? Question: How to establish an explicit characterization of both the optimal value functions and the optimal control policy for a controlled random walk? Background: Assume our system is a perfectly-... 6 votes 1 answer 87 views ### Minimize \int_0^\infty g'(x)f(x)\,dx where f(x) has a log-normal density I would like to optimize (minimize) the following expression in order to find the functional minimizer g (which should be at least once differentiable): \int_0^\infty g'(x) f(x) \ dx $$where f(... 2 votes 0 answers 60 views ### FOC point vs Stationary point in local optimization In this SIAM Review paper the authors are giving the following necessary condition for a point being a local maximum of a convex function: Let F: \mathbb{R}^n \mapsto \mathbb{R} be convex. If x ... 5 votes 1 answer 375 views ### Optimality in L Shaped or Bender Decomposition I was working on solving a two-stage stochastic problem using L Shaped method (Benders Decomposition). I have discussed the model here: Stochastic Facility Location Model. Do the single-cut/ multi-cut ... 2 votes 1 answer 146 views ### Promising regions in optimization I have investigated the literature, but I could not find proper explanations. As we know, (meta-)heuristics try to explore promising regions of optimality. The regions are then exploited. What is the ... 4 votes 1 answer 85 views ### How to prove pseudo-convexity of a discrete function? Given a general function f:\Bbb Z\to\Bbb R is there a simple way to verify whether f(x) is pseudo-convex or not? 8 votes 3 answers 2k views ### How to determine different gap rates? I found in the literature different gaps: a gap between a random solution and an exact solution a gap between the exact solution and a lower bound a gap between the exact solution and a lower bound a ... 0 votes 1 answer 2k views ### Is optimal solution to dual not unique if optimal solution to the primal is degenerate? If optimal solution to the primal is degenerate, does it necessarily follow that optimal solution to dual not unique? That is, is uniqueness an unnecessary assumption? Spin-off from here. In my ... 11 votes 1 answer 304 views ### Finding primal feasible solution from optimal dual I'm reading Boyd's notes on forming the dual problem in order to decompose the primal problem. On page 4, right before the start of the next section, he talks about how given the optimal dual solution,... 5 votes 2 answers 179 views ### Local optimum of dual of non-linear program In general, suppose you have a non-convex optimization problem with constraints and you form the dual problem. If you find a local optimum for the dual problem, will the corresponding primal solution ... 6 votes 2 answers 700 views ### Existence of Optimal Solution Assume we are solving \min\{f(x) \ | \ x \in S \}. If f: \mathbb{R}^n \mapsto \mathbb{R} is a proper closed convex function, and S is a non-empty closed convex set, does this imply that the ... 8 votes 2 answers 194 views ### Conditions for minima in calculus of variations In the calculus of variations (unconstrained), one applies a first-order variation on a general functional of the form$$\int_{a}^{b}F(x,y,y')\,dx to obtain the first-order necessary condition for ...
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It seems like most solvers that can tackle nonlinear nonconvex optimization problems (e.g. IPOPT) operate on ultimately solving for the first-order optimality conditions. Can it therefore be assumed ...