Questions tagged [optimality-conditions]
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17
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How to check optimality of conic optimization problem
I'm trying to solve this problem, but I'm not sure if it is possible to check the optimality of this problem.
$$\min_{K,L} \quad Tr(L^\top L)\qquad\\
\text{s.t.} \quad K^\top L = A^\top Q\\
\qquad \...
2
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1
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167
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Difference between Optimality cuts and Feasibility cuts for L shaped method in stochastic programming?
What is the difference between Optimality cuts and Feasibility cuts for L shaped method in stochastic programming? Like for the following problem they used Optimality cuts,
$$
\begin{aligned}
& z=\...
- 157
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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-...
- 141
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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(...
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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 $x$ ...
- 3,980
5
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1
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387
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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 ...
- 629
2
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1
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146
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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 ...
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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?
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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 ...
- 413
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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 ...
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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,...
- 329
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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 ...
- 329
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2
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769
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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 ...
- 3,980
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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 ...
- 715
11
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556
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Solvers and saddle points
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 ...
- 715
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Optimality in a simultaneous column and row generation procedure
What is the optimality argument in a simultaneous column and row generation procedure? By column and row generation procedure I mean a procedure in which every time a column in generated, several ...
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Global optimality condition of non-convex quadratic programs
We know that a convex quadratic maximization (not minimization!) on a polyhedron has its global optimal value on a vertex.
Also, I have read in some papers that checking whether a vertex is globally ...
- 3,980