# Questions tagged [non-convex-optimization]

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### Minimising kurtosis (non-convex). Can I use algebraic geometry or alternate methods to show uniqueness of a particular solution?

I consider a weighted sum of $n$ identically-distributed correlated random variables. The weights in the sum, $w_i$ for $i=1, 2,...,n$, satisfy $w_i>=0$ and $\sum_{i=1}^{n}w_i=1$. I am ...
154 views

### Nonconvex optimization with linear constraints

Which algorithms are suitable for solving problems of the form $$\min_x \lbrace f(x) \; | \; Ax \leq b \rbrace$$ with nonconvex, differentiable obfective $f$. Unfortunately, $f$ cannot be assumed to ...
1 vote
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### If $\ell_0$ regularization can be done via the proximal operator, why are people still using LASSO?

I have just learned that a general framework in constrained optimization is called "proximal gradient optimization". It is interesting that the $\ell_0$ "norm" is also associated with a proximal ...
887 views

### Solving a linear program, but over the unit sphere

I want to solve a linear program but with a subset of the variables taken from a unit sphere. That is, given fixed $\textbf{c} \in \mathbb{R}^{n}$, $\textbf{A} \in \mathbb{R}^{m \times (n+k)}$, I want ...
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### Quartic optimization problem over the unit Euclidean sphere

I want to solve following optimization problem in $x \in \mathbb R^n$. $$\begin{array}{ll} \text{maximize} & \displaystyle\sum_i(x M_i x^T)^2\\ \text{subject to} & \|x\|_2 = 1\end{array}$$ ...
58 views

### Solving Mixed-Integer Non-Linear Optimization Problem

I would like to solve the following optimization problem: \begin{array}{ll} \underset{x_{i}\geq0,\, \pi_{i}\in\{0,1\}}{\text{minimize}} & \displaystyle\sum_{i=1}^n x_i\\ \text{subject to} & ...
258 views

### Program to solve Optimization Problem

I have an optimization problem, this problem has linear constraints and nonlinear constraints. I solved the linear part by MATLAB but the nonlinear constraints I could not solve it. I downloaded ...
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4k views

### Maximizing a convex function with a convex constraint

Given a convex function $f : \mathbb{R}^n \to [0,\infty)$, the objective is to find the farthest point in the level set $\left\lbrace x \in \mathbb{R}^n \mid f(x) \leq 1\right\rbrace$ (Assuming that ...
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### Can you give me good examples of non-convex functions that are problematic for optimization?

I want to test my extended gradient descent algorithm, whose aim is to handle non-convex problems better. Can you give me some examples of non-convex functions that are hard to minimize via gradient ...
1 vote
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5k views

### Linearly constrained eigenvalue problem

Suppose I'd like to: \begin{align} \mathop{\text{min}}_\mathbf{x} && \mathbf{x}^T\mathbf{A}\mathbf{x} \\ \text{subject to:} && \mathbf{x}^T \mathbf{M} \mathbf{x} = 1\\ && \...
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### Maximizing quadratic form on the hypercube

I want to maximize a quadratic form $\mathbf x^T\mathbf Q\mathbf x$ and also want to find out which vector $\mathbf x$ maximizes the quadratic form when $\mathbf Q$ is an $n\times n$ positive ...