# Questions tagged [convexity]

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

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### Find a dual problem with one dual decision variable to the problem of finding the orthogonal projection of a given vector

Given the set $T_{\alpha}=\{x\in\mathbb{R}^n:\sum x_i=1,0\leq x_i\leq \alpha\}$ For which $\alpha$ the set is non-empty? Find a dual problem with one dual decision variable to the problem of finding ...
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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 ...
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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 ...
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### Convexity of a QP

In quadratic programming (QP), you encounter an objective of the following form: $$x^TQx + c^Tx$$ and often it's desirable to know if the QP is convex. One method to check for convexity is by ...
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I'm looking for examples of (classes of) problems with a non-convex, non-linear formulation, but convex feasible region. That is, a problem of the sort: \begin{array}{lll} \text{minimize} & ... 2answers 485 views ### 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 ... 3answers 3k views ### 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 ... 1answer 340 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 ... 5answers 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 ... 2answers 340 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 ... 1answer 470 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 ... 2answers 249 views ### How to determine the convexity of my problem and categorize it? My problem is: \begin{align}\min\limits_{x_{ij}}\qquad&{\sum_{i\in N}\sum_{j\in M}\frac{x_{ij}}{C_j-\sum\limits_{i\in N} x_{ij}a_i}}\\\text{s.t.}\qquad&0<C_j-\sum_{i\in N} x_{ij}a_i\\\qquad&... 1answer 216 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 \begin{equation} x_i = ... 2answers 379 views ### How can I linearize or convexify this binary quadratic optimization problem? I have an optimization problem as below. I am having a hard time with the last constraint. \max \eta subject to {\bf U}(:,m)^T{\bf A}{\bf U}(:,m)=0,m=1,2,\cdots,M here \bf{A} is a Binary ... 2answers 148 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, ... 1answer 377 views ### KKT inequality conditions Let's say I have an objective functionf(x_1,x_2, \cdots, x_n)$$and N constraints$$x_i \ge 0. $$I am trying to solve it with KKT conditions. Now the objective function becomes$$f(x_1,x_2,...
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 ...