# Questions tagged [non-convex-optimization]

A non-convex optimization problem is one where either the objective function is non-convex in a minimization problem (or non-concave in a maximization problem) or where the feasible region is not convex.

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### C-VaR approximation problem

In the paper written by Rockafellar about C-VaR (https://www.ise.ufl.edu/uryasev/files/2011/11/CVaR1_JOR.pdf), it is explained that this quantity can be approximated using the following problem (...
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### Proof of convergence for a heavy-ball adaptive step-size algorithm for non-convex functions

I am struggling with prooving convergence for an optimizer which uses adaptive step-size with heavy ball algorithm for convex and non-convex functions. In some literature, I could find a regret bound ...
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### DC programming and saddle-point problems

Is there any relation between two? DC programming problems $$\min_{x\in \mathbb{R}^n} f(x)-g(x),$$ where $f,g$ are both differentiable and convex functions. Saddle-point programming aim to solve the ...
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### given a binary point, design a quadratic which is minimized at that point!

Given a binary vector $x$, I need to efficiently construct a matrix $A$ such that $x$ is a global minimizer of $z^TAz$ over binary $z$'s. I need the diagonal elements to be positive.
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### is a function which all stationary points are global minima prox-regular for some $\alpha$>0?

Let $f$ be a function which all stationary points are global minima. This type of functions are also known as invex. It means, there exits $\eta(x,y)$ such that $f(x)-f(y) \geq \zeta_{y}^{T}\eta(x,y)$ ...
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### Box-constrained QCQP

Let ${\bf A} \in \mathbb{R}^{n \times n}$ be a symmetric positive semidefinite (PSD) matrix, let ${\bf a} \in \mathbb{R}^n$ and let ${\bf B} \in \mathbb{R}^{n \times n}$ be a symmetric indefinite ...
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### convert constrained problem to unconstrained problem

I am currently working on a mathematical problem involving a non-convex function, $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$. I have a constrained optimization problem that I would like to convert ...
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Consider the optimization problem: \begin{aligned} \text{maximize}&\sum_{k=1}^9\left(x_k+\frac{k}{9}\right)^2\\ \text{subject to}&\begin{cases}\bar{x}\succeq \bar{0}\\\sum_{k=1}^9x_k=1.\end{... 0 votes 1 answer 85 views ### Linear program plus sphere constraint I have the following optimization problem: \begin{align} \min_{x} \quad & x^\top \hat{x} \\ \text{s.t.} \quad & x^\top a_i \leq 0, \quad \forall i \in \{ 1,\ldots,N \} \\ & \|x\|_2 = 1, \... 0 votes 0 answers 13 views ### Proving lower bounds on a minimization problem over positive semidefinite matrices with a bounded maximal rank Given a minimization optimization problem of a linear target function over the set of positive semidefinite matrices of some fixed maximal rank, subject to affine constraints, what are (analytical) ... 2 votes 2 answers 86 views ### Minimum of a non-convex function Let, \begin{align} f(x,y)=\frac{1}{y^2}+\left(\frac{x}{y}+\sqrt{2-\frac{1-x^2}{y^2}}\right)^2, \end{align} where 0 \le x\le 1, -1 \le y \le0, x^2 + y^2 \le 1 and x^2 + 2y^2 \ge 1. What is the ... 0 votes 1 answer 21 views ### Convexity of set of mixture distributions A set of mixture distributions Q is defined as, Q = \{f | f(.) = \sum_{i=1}^{k} q_{i} f_{i}(.) \}, where each f_{i} is a probability density and \sum_{i=1}^{k} q_{i}=1 and q_{i}>0. Is set ... 0 votes 1 answer 81 views ### Optimal solution to the primal and the Lagrangian dual when there is duality gap Consider the primal problem \max_{x\in\mathcal{X}} f(x) subject to g(x)\leq 1 and its Lagrangian dual \min_{\lambda\geq 0}\max_{x\in\mathcal{X}}\{f(x)+\lambda(1-g(x)\} with no convexity ... 0 votes 0 answers 25 views ### When do equality and inequality constrained problems agree? Let f:\mathbb{R}^d\to\mathbb{R} and h:\mathbb{R}^d\to\mathbb{R}. Consider a solution x^\ast to the equality constrained optimisation problem x^\ast \in \underset{x}{\text{argmin}} f(x) \quad\...
I am trying to formulate an optimization problem with the following constraint: $y = 1$ if $x \le c$ and $y = 0$ if $x > c$ which is basically an indicator function $y = 1[x \le c]$ and $c$ would ...