Questions tagged [non-convex-optimization]

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23 views

Under what condition can we prove $\nabla_x \min_y f(x,y)=\nabla_x f(x,y^*)$ where $y^*=\arg\min_y f(x,y)$?

Let $f: \mathbb R^n\times \mathbb R^m\to \mathbb R$ be a function. I wonder under what condition can we prove $\nabla_x \min_y f(x,y)=\nabla_x f(x,y^*)$ where $y^*=\arg\min_y f(x,y)$. For example, ...
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39 views

Minimize smooth function $(x,y) \to f(x,y)$ subject to $x \perp y$

Let $V$ be a finite-dimensional real vector space (e.g space of $m \times n$ real matrices equiped with Hilbert-Schmidt inner product $(A,B) \to \mathrm{tr}(AB^\top)$, and let $f:V^2 \to \mathbb R$, $(...
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1answer
54 views

PCA, relation between the error and variance

As is known, the rank-1 PCA aims to solve the following optimization problem $$\min_{x\in\mathbb{R}^d}\quad -x^T \Sigma x\quad\quad\quad \text{s.t.}\quad \Vert x\Vert_{2}=1,$$ where $\Sigma\in\mathbb{...
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0answers
19 views

Understanding non-convex subgradients and normal cones

I think I have a very good understanding of subgradients of convex functions and normal cones to convex sets. On the other hand, I have a lot of difficulties understanding them in the non-convex setup....
2
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0answers
47 views

Global minimum of sum of a non-convex and convex function, where minima of the non-convex function can be found

I'm interested in finding $\arg\min_{x \in X} (f(x) + \lVert x\rVert_2^2)$ where $X$ is a $[0,1]^n$, $f$ is Lipschitz but non-convex and we already have a procedure to find some $x^* \in \arg\min_{x\...
1
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1answer
56 views

Prove zero slope point is global maximum for constrained function with binomials. Without restriction, objective function is non-concave

How to prove the zero slope point is a global maximum in this non-concave program for a function with binomials? I need to find the (global) maximum of the following constrained problem: $$\max_{CAP} \...
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38 views

A parametrized saddle point problem with linear constraints

I am struggling to find any potential algorithm for solving a saddle point problem. More precisely let $\mathcal{P}=\{ \mathbf{x}\in \mathbb{R}^{d}; \mathbf{A}\mathbf{x}=\mathbf{b}, \mathbf{x} \geq 0\}...
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0answers
46 views

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 ...
3
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1answer
46 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 ...
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159 views

Minimizing the Frobenius norm of a quadratic matrix expression

Given matrices $R \in \mathbb R^{m \times n}$ and $Y \in \mathbb R^{p \times n}$, where $R$ is full rank, how can I solve the following optimization problems? $$\min_{X \in \mathbb R^{p \times m}} \| ...
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87 views

Derivation of general primal and primal-dual interior point method and their differences

I am studying interior point methods for general nonlinear optimization. We have a problem with $n$ decision variables, $m_e$ equality constraints and $m_i$ inequality constraints. We assume that $f(\...
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47 views

Why the result of the non-convex optimization problem will be farther and farther away from the optimal

When I try to solve a optimization problem by Riemannian stochastic variance reduced gradient algorithm(RSVRG), the formulation of problem like $\frac{1}{N}\sum_{i=1}^Nf_i(x)$ and $f_i(x)$ is a non-...
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42 views

(Iterative?) Solutions to a certain quadratic program with non-convex constraints

Let $y\in\mathbb{R}^m$, $\tau\in\mathbb{R}$ and $X\in\mathbb{R}^{m\times n}$, with $\tau>0$ I would like to efficiently solve the following problem: Problem 1 Choose $\alpha,z\in\mathbb{R}^m,\beta\...
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102 views

Convergence of heavy-ball method for non-convex optimization

The heavy-ball method (also called gradient descent with momentum) is commonly used in optimization. The update rule can be written as: $$x_{t+1}=x_t-\eta\nabla f(x_t)-\beta (x_t-x_{t-1})$$ Suppose $\...
2
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1answer
196 views

Hardness of concave minimization problem

I have an optimization problem $\underset{x}{\min} ~ c(x) - k \cdot x$ where $c(x)$ is a non-decreasing concave function with $c(0) = 0$, $x \in C \subseteq \mathbb{R}^d_{\geq 0}$. By non-decreasing, ...
2
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0answers
55 views

Variational forms of non-convex functions

I am trying to understand what kind of variational forms exist for non-convex functions. Alternatively, are there conjugate forms which attain strong duality? For a non-convex function $f$, I am ...
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0answers
123 views

Necessary optimality condition for quadratic programming: a solution of a constrained QAP is a solution of a LP

I have a concern about a result given by Murty in [1] and also written by Floudas and Visweswaran in [2] They consider a QP: \begin{array}{ll}{\min _{x} Q(x)} & {=c^{T} x+\frac{1}{2} x^{T} D ...
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1answer
176 views

A quadratic program with non-negativity constraints

Is there any closed form solution for the optimal value of the folowing optimization problem? $$\begin{array}{ll} \text{minimize} & (\mathbf{x} - \mathbf{y})^{\mathrm{T}}\mathbf{B}(\mathbf{x} - \...
5
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1answer
281 views

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 ...
5
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1answer
419 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 ...
2
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0answers
150 views

Quartic optimization problem over the unit Euclidean sphere

I want to solve following optimization problem in $x \in \mathbb R^n$. $$\begin{array}{ll} \text{maximize} & \displaystyle\sum_i(x M_i x^T)^2\\ \text{subject to} & \|x\|_2 = 1\end{array}$$ ...
2
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0answers
54 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} & ...
3
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2answers
170 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 ...
7
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2answers
354 views

Proving an infinite norm minimization problem has finite support (non-convex p-norms)

Consider an optimization problem over infinite variables: $$ \begin{align} \min_{x}~& {\left\lVert{x}\right\rVert }_p \\ \text{s.t}~& \left\langle x, a_n\right\rangle \ge 1~,~\forall n=1,\...
4
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1answer
196 views

Optimizing a multivariate symmetric (permutation-invariant) function

Let $\ell$ and $d$ be two integers such that $\ell \le d$. I would like to find the global maxima of the following symmetric function $f\colon (0,1]^n \to \mathbb{R}$, $$f(x_1, \ldots, x_n) := \sum_{\...
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2answers
2k 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 ...
2
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1answer
79 views

Linear optimization with one positive definite quadratic equality condition in P?

I have the following minimization problem in $z \in \mathbb R^n$, which contains $x_1, \dots, x_t, y \in \mathbb R$. $$\begin{array}{ll} \text{minimize} & y\\ \text{subject to} & xQx'= y\\ &...
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0answers
143 views

non-convex optimization with constraint

I have a special non-convex optimization problem: $\min / \max \ f(x) + g(x) + h(x)$, subject to $| g(x) - h(x)| < \varepsilon$, where $f(x)$ is non-convex, but both $g(x)$ and $h(x)$ are ...
1
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1answer
309 views

Maximizing quadratic form subject to inequality constraints [closed]

Given a $n \times n$ symmetric matrix $\rm S$, solve the optimization problem in $n \times k$ (where $n \geq k$) matrix $\rm X$ $$\begin{array}{ll} \text{maximize} & \mbox{tr} \left( \mathrm X^\...
7
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2answers
364 views

Eigenvalue problem with two quadratic constraints

I would like to solve the following problem: $$\begin{array}{ll} \text{minimize} & \mathbf{x}^T \mathbf{A} \mathbf{x}\\ \text{subject to} & \mathbf{x}^T\mathbf{B}\mathbf{x} = 0\\ & \...
3
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1answer
86 views

Why to multiply the penalty by $n$ in the penalized least squares and likelihood?

In the SCAD paper by Fan and Li (2001), there exist two forms of penalized least squares as follows: $$\frac{1}{2}\left \| y-X\beta \right \|^2+\lambda \sum_{j=1}^{d}p_j (\left | \beta _j \right |),$$ ...
1
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1answer
277 views

Are there any solvers to Chance Constrained Programming Problems

I'm trying to solve a chance constrained programming (CCP) problem $\min_x f_0(x, \xi), \text{ such that } \mathbb{P} ( f_i(x, \xi) \ge \alpha_i ) \le \epsilon_i, \text{ where } i = 1,2,\cdots, m$ ...
5
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1answer
318 views

Maximize the determinant of Boolean combinations of positive definite matrices

I have the following optimization problem. $$\begin{array}{ll} \text{maximize} & \det \left(\sum^n_{i=1}z_i W_i \right)\\ \text{subject to} & \sum_{i=1}^n z_i = N\\ & z_i \in \{0,1\}\end{...
4
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1answer
3k views

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 ...
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0answers
243 views

Is this QCQP convex or nonconvex?

\begin{equation} \begin{split} \min_{x\in \mathbb{R}^n}\:f(x)=(1/2)x^{T}Q_0x+c_0^T x \end{split} \end{equation} s.t. $$ g_i(x)=\frac{1}{2}x^T Q_ix-lmax_i\leq0,i\in\{1,...,m/2\} $$ $$ g_i(x)=\frac{...
5
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2answers
2k views

Quadratically constrained linear program (QCLP) over $x$ with the linear constraint $x = Az$

I have a problem that looks very much like a norm-constrained linear program, but with an extra constraint that is unusual for me. The problem is the following. Given a matrix $A$ and a vector $w$, $$...
12
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2answers
4k 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\\ && \...
5
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1answer
1k views

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 ...
3
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1answer
683 views

Nonconvex optimization problem

I have a nonconvex optimization problem with a linear objective function, a set of linear constraints and a set of nonlinear, non-convex constraints. Is this problem NP-hard? If so, how can I prove ...
3
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1answer
398 views

Is the feasibility of a system of non-convex quadratic equations and inequations decidable?

I would like to know whether the following problem is decidable. Given the following system in $x \in [0,1]^n$ $$x^T Q_i x + r_i = 0 \mbox{ for } i = 1, ..., k$$ $$x^T Q_j x + r_j \neq 0 \mbox{ ...
2
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1answer
161 views

Non-convex quadratic optimization

I would like to optimize the following system: $$\min_{q,\|q\|=1} \sum_i^n |q^T M_i q|$$ More details: the size of the unknown vector $q$ is $4 \times 1$, $M_i$ is a matrix of size $4\times 4$. It ...
1
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1answer
611 views

A non-convex quadratically constrained quadratic program

$$\begin{array}{ll} \text{minimize} & \beta^{T} A \beta\\ \text{subject to} & \beta^{T} C \beta=1\\ & \beta \geqslant 0\end{array}$$ where $A, C\in \mathbb{R}^{M\times M}$ and $\beta \in ...
2
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1answer
1k views

Can one maximize the spectral norm of a matrix via semidefinite programming?

Consider the following optimization problem: Maximize $\|X\|_2$, subject to $X$ being Hermitian (or symmetric) and a bunch of semidefinite constraints on $X$. Here, $\|X\|_2$ is the spectral norm of ...
2
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3answers
2k views

Solving a non-convex quadratically-constrained quadratic program

I have the following quadratic optimization problem: $\min_{\vec{x}} |\vec{x}|^2$ subject to $\vec{x}^T G_j \vec{x} \geq 1$, $j = 1 \ldots m$, where the $G_j$ are positive semidefinite. $|\vec{x}|$ is ...
2
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2answers
3k views

Projected gradient descent for non-convex optimization problems

My question is in regards to the minimization of a convex function where the feasible set of solutions is non-convex. Can projected gradient descent (PGD) be used here to obtain a stationary solution? ...
48
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7answers
22k views

Is all non-convex optimization heuristic?

Convex Optimization is a mathematically rigorous and well-studied field. In linear programming a whole host of tractable methods give your global optimums in lightning fast times. Quadratic ...