Questions tagged [optimality-conditions]

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5
votes
2answers
83 views

how to determine differents gap rate?

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
1answer
151 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
0answers
123 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,...
6
votes
2answers
103 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
2answers
145 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
2answers
160 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 ...
11
votes
1answer
423 views

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 ...
18
votes
1answer
129 views

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 ...
10
votes
2answers
137 views

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 ...