# Questions tagged [convexity]

For questions related to convex functions and convex sets, especially as they relate to optimization problems.

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### Reference for “expectation preserves convexity”

It is well known that expectation preserves convexity: If $f(x)$ is convex and $Y$ is a random variable, then $\mathbb E[f(x-Y)]$ is convex. This property arises in, for example, inventory theory. I ...
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### Can an integer optimization problem be convex?

I'm trying to wrap my head around an apparent paradox that I've come across while trying to learn more about optimization algorithms: On one hand several sources state that convex optimization is ...
3k views

### Linear Programming with additional “if-then”/“Default to zero” constraints?

What approaches can I use for a Linear Programming problem with the additional constraint that if a decision variable falls below a certain threshold, then it should just be forced to 0. I'm ...
136 views

### Bound on the number of constraints to be generated (lazy constraints)

I am working on a very large optimisation problem. All variables are continuous, the objective is linear and the constraints convex, but I have many such constraints (on the order of $2^n$ — actually, ...
283 views

### Convexity of Variance Minimization

$X$ is a discrete random variable taking value $x_n$ with probability $1/N$ for $n=1, \ldots,N$. I would like to set the $x_n$ values in an optimization problem. My objective is to minimize the ...
438 views

### How to formulate a problem to prove/disprove convexity?

Given a general non-linear problem: \begin{align}P:\qquad&\min_{x\in X} f(x)\\\text{s.t.}\qquad&g(x)\leq 0\end{align} where $f$ is a non-linear function, $g$ is a vector of non-linear ...
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### Is This Constraint Convex?

I have a constraint that I believe to be convex and not affine which I think means that I can implement a relaxation. I will first define the full constraint, and then build up my (informal) reasoning ...
275 views

### Convex vs Strictly Quasiconvex Functions in Optimization

I have read/heard quite a few time that in the old days, it was considered that linear programs constitute the class of optimization problems that can be solved efficiently in practice (as a rule of ...
418 views

### Convex Maximization with Linear Constraints

I am doing active research in convex maximization w.r.t. linear constraints. There are many cases which can be efficiently approximately solved, e.g., convex quadratic maximization, log-sum-exp ...
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### Dedicated solver for convex problems

Are you aware of a fast solver (open source or commercial) for convex NLPs that is faster than IPOPT? I'm interested in problems in the 50K+ variable range, both dense and sparse. Ideally, it would be ...
181 views

### Recovering primal optimal solutions from dual sub gradient ascent using ergodic primal sequences

My question concerns recovering a primal optimal solution while performing dual sub gradient ascent. Denoting by $y_i$ the dual multiplier in the $i^{\text{th}}$ iteration, let x_i = ...
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### Problem solvable $\Rightarrow$ subproblems solvable if feasible region closed, convex?

Let $c \in \mathbb{R}^n$, $M \subseteq \mathbb{R}^n$ such that the problem \begin{align}P:\quad\min_{x \in \mathbb{R}^n}&\quad c^\intercal x\\\textrm{s.t.}&\quad x \in M\end{align} is solvable....
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### Solving convex programs defined by separation oracles?

General question: What software can solve convex programs defined by a separation oracle? The objective function is concave, and the feasible set is a polytope. By a separation oracle I mean that I ...