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.

# Questions tagged [non-convex-optimization]

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### Can metaheuristics be used to optimize a non-convex function $f(x_1,x_2,y_1,y_2) = \frac{y_1}{x_1} + \frac{y_2}{x_2}$?

I want to use genetic algorithm for minimizing the above function where $x_1,x_2,y_1,y_2 \in \mathbb{R}$. Is it possible to find a local/approximate solutions using metaheuristics such as genetic algorithms? Or what are the methods can we use to…

thesukantadey

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### Find matrices $X$ such that the diagonal of $X^H X$ is sparse

I want to find matrices $X$ such that the diagonal of $X^H X$, where $X^H$ is the Hermitian transpose of $X$, is sparse — as many zeros as possible. Intuitively, $X$ should have many empty columns.
I guess this is one of the sparse matrix recovery…

Disenchanted Toad

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### Literature on optimizing linear objectives over non-convex sets

I was trying to find related literature on linear optimization problems with a non-convex constraint set. Could you provide links to the related literature? Thank you.

Alex

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### Solution of a semidefinite program with rank at most $k$

I'm working on a semidefinite program (SDP) in $X \ge 0$, and I want to obtain an optimal solution $X^*$ with rank at most $k$, where $k$ is typically a small integer.
Let $0 \le \lambda_1 \le \lambda_2 \le \cdots \le \lambda_n$ be the eigenvalues…

StevenG

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### Optimization of the sum of a convex and a non-convex function?

I have two quadratic functions $f(x)$ and $g(x)$ which are convex (PSD Hessian)and non-convex respectively and $x\in R^{n}$.
Adding them together as $(1-\lambda)f(x)+\lambda g(x)$ for $\lambda=\lambda^*$ their sum will become convex (PSD Hessian).…

Bob

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### How should I proceed to solve the below mentioned non-convex optimisation problem?

I am a newbie in the optimization domain. I am trying to solve a real-life problem. Where I have a set of equations to solve, with multiple variables.
The main equation looks like
$x1[(w1*w7)+(w4*w8)]+x2[(w2*w7)+(w5*w8)]+x3[(w3*w7)+(w6*w8)] =…

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### Can someone provide an example why/how the Hyperplane Separation Theorems don't hold for non-convex sets?

I understand intuitively that they won't hold, but I can't find any good examples.

Drew B.

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### Difference of convex functions for $x/y^2$

Let $f=\frac x{y^2}$ be a positive function defined on the domain $x>0$ and $y>0$.
I want to express $f$ as $g-h$ where both g and h are convex.
Is there an easy way to do that?

Wazowski

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### convexity of $\frac{ \vec{a}^{\top}\vec{b}} {\vec{a}^{\top}\vec{c}}$?

I want to add $f(\vec{x})=\frac{ \vec{x}^{\top}\vec{b}} {\vec{x}^{\top}\vec{c}}$ to an optimization problem as an additional objective, such that:
$$\min_x g(\vec{x})+f(\vec{x}) \\
s.t ~~x_i \ge 0$$
in which $g(x)$ is convex, and $\vec{b}$ and…

Bob

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### Optimization over unit vectors

I am trying to optimize a certain nonlinear function over real or complex vectors with unit $L^2$ norm, i.e., I am trying to maximize a function $|f(x_1,\ldots,x_n)|^2$ subject to $\sum_i |x_i|^2 =1$. I am wondering if this goes by a special name or…

Nicolás Quesada

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### How could I optimize the objective function which is a probability form

Recently, I am trying to solve a stochastic planning problem. The problem is like this:
$$
\max_{x\in \mathbb{R}^n,x\geq0}\quad \varphi(x) = \mathbb{P}[\xi\leq b+Ax] \\ Cx\leq d
$$
In this problem, $Cx\leq d$ is linear constraints, which is not a…

X.Li

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### Linearization of two continuous variable division

I want to linearize the following function
$f(x)=\frac{\theta}{2(1-\rho)}$
where, $\theta$ and $\rho$ are continuous variables. When I checked the convexity, I found that it is not convex since one of the eigenvalues is positive and the other is…

tcokyasar

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### Matrix completion by rank minimization

In matrix completion, the starting point is often stated as: the optimization problem for matrix completion:
$$\begin{array}{ll} \text{minimize} & \frac 12 \| X - M \|^2\\ \text{subject to} & \mbox{rank} (X) \leq r\end{array}$$
where $X$ is the…

val

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### Frank Wolfe algorithms for non-convex optimization

I have a non-convex function as follow
image
where: $\alpha < 1$ and $\theta \in \bigtriangleup_{K}$ means:
$\theta_1 + \theta_2 + ... + \theta_K = 1$ and $\theta_i \ge \epsilon > 0, \quad i=1,...,K$ ($\epsilon$ is a small constant)
I design a FW…

Khang Truong

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### How to solve this nonlinear constrained optimization efficiently?

Set $\{\alpha_i\}$ and $\{\beta_i\}$ as the variables, the optimization problem I need to solve is as follows:
\begin{equation}
\begin{split}
\min\max_{i}\quad &\frac{(a_i^2+b_i^2+c_i^2+d_i^2)^2}{(a_id_i-b_ic_i)^2}
…

Chenfl

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