# Questions tagged [optimality-conditions]

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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?
103 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 ...
249 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 ...
160 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,...
110 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 ...
161 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 ...
163 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 ...
440 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 ...