# Questions tagged [nonlinear-optimization]

Nonlinear objectives, nonlinear constraints, non-convex objective, non-convex feasible region.

233 questions with no upvoted or accepted answers
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### $L_2$ minimizing makespan vs. $L_\infty$ minimizing makespan

There are $n$ positive real numbers. We partition these numbers into $m$ parts, the size of each part is the sum the numbers in this part. Maximum size of the parts is called a makespan of a partition....
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### Solving matrix equation $X^{-1}=\sum_{i=1}^n D_i X A_i$

Does anybody know an algorithm to solve the following matrix equation? $$X^{-1}=\sum_{i=1}^n D_i X A_i$$ where $D_i$s are diagonal and $A_i$s are symmetric matrices. It would be great to have an ...
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### Concavity of a function implicitly defined by a polynomial

Consider the following system of $n$ equations: \begin{equation}f_j^2 = x_j^2\sum_{i=1}^n A_{ij} f_i \tag{$\star$} \end{equation} where $A_{ij}\geq 0$ are known constants and where $x_j>0$ for ...
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### Bounding volume of cell in complement of zero set

I am given an integer polynomial $f \in \mathbb{Z}[X_1, \ldots, X_n]$ of bounded degree and bounded coefficient size. The polynomial's zero set partitions $\mathbb{R}^n$ into cells. What I am looking ...
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### measure of an image under an argmax function

I am trying to find any techniques to analyze the measure of an image of a set under an argmax function. For example, let $\Omega\subset\mathbb{R}^n$ be compact and $\phi:\Omega\to\mathbb{R}$ be ...
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### Find subset of collection of sets whose intersection has minimum average value

Let $a_1,\ldots,a_n>0$, and let $S_1,\ldots,S_d\subset\{1,\ldots,n\}$ (all non-empty). For any $I\subseteq\{1,\ldots,d\}$, define $S(I)=\bigcap_{i\in I} S_i$. Given some $1\leq s < d$, consider ...
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### Minimizing the largest eigenvalue of matrix product

Let $A\in\mathbb{C}^{m\times n}$, $B\in\mathbb{C}^{n\times k}$, $C\in\mathbb{C}^{k\times m}$ be given complex matrices. The objective of the optimization problem is \begin{equation} \mathop {\arg \min ...
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### A conjecture about the barycenter of a polytope

Could someone help me with the following conjecture? Thanks a lot! Suppose I have a polytope $\Delta$ in $\mathbb R^n (n\geq 2)$ with coordinates $(x_1,x_2,\cdots,x_n)$ defined by linear ...
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### Numerical scheme for convex optimization

Given $(e_n)_{-N\le n\le N}\in\mathbb R^{2N+1}$ and $-1<x<1$, solve \begin{eqnarray} &&\max_{(q_n)_{-N\le n\le N}\in\mathbb R^{2N+1}_+}~ \sum_{n=-N}^N (e_n-\log(q_n))q_n \\ \mbox{s.t.} &...
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### An inequality for three iid random variables with a log-concave density

It was previously shown that $$H\ge cG,\tag{1}$$ where $c:=1/14334$, $$G:=E|X-Y|,\quad H:=E|X-Y|-\tfrac12\,E|X+Y-2Z|,$$ and $X,Y,Z$ are independent random variables with the same log-concave density. ...
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### Convex optimization upper bound for a non-linear optimization

Is there any good convex optimization problem based upper-bound for the following non-linear optimization problem? \begin{align} \max_{x_1,\ldots,x_N}&\quad \sum_{n=1}^{N} \log(1+\frac{x_n}{1+\...
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### Convergence of iteration of a convex program

Let $\mathbf{V} \in \mathbb{R}_{+}^{n \times m}, \ \ \mathbf{E} \in \mathbb{R}_{+}^{n \times m}$, with $\mathbf{V} \mathbf{1}_{m} = \mathbf{1}_{n}$ and $\mathbf{E}^{T} \mathbf{1}_{n} = \mathbf{1}_{m}$...
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### How can we solve this kind of saddle point problem?

I'm trying to solve a saddle point problem of the following form: Let $(E,\mathcal E,\lambda)$ be a measure space; $p$ be a probability density on $(E,\mathcal E,\lambda)$ and $\mu:=p\lambda$ $W$ be ...
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### Maximize an $L^p$-functional subject to a set of constraints

Let $(E,\mathcal E,\lambda)$ and $(E',\mathcal E',\lambda')$ be measure spaces $f\in L^2(\lambda)$ $I$ be a finite nonempty set $\varphi_i:E'\to E$ be bijective $(\mathcal E',\mathcal E)$-measurable ...
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### Looking for a very particular kind of non-convex functions

I want some examples (hopefully parametric families!) of non-convex functions which satisfy the following properties simultaneously, It should be at least twice differentiable. It should have a ...
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### Can a quadratic matrix inequality constraint be convex?

I have an optimization problem with a semi-definiteness constraint: $$N \preceq 0$$ where the entries $N^{AB}$ of the matrix $N$ are defined through $$N^{AB} = \sum_{i,j} x^i M_{ij}^{AB} x^j$$ The ...
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### Are there scenarios under which feasibility bilinear programming is easy?

Given $c\in\Bbb R^{n_1},d\in\Bbb R^{n_2}$, $E\in\Bbb R^{n_1\times n_2}$, $A\in\Bbb R^{m_1\times n_1}$, $B\in\Bbb R^{m_2\times n_2}$ $a\in\Bbb R^{m_1}$, $b\in\Bbb R^{m_2}$ and $t\in\Bbb R$ we know ...
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### Condition under which the Clarke's subdifferential is locally Lipschitzian

Given a locally Lipschitz continuous function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ and closed set $$\Omega =\left\lbrace x \in\mathbb{R}^n \ |\ f(x) \leq 0 \right\rbrace$$ such that f is semi-...
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### Does this non-negative function, with No stationary points, have only descend directions close to a constraint set?

Suppose $P: \mathbb{R}^n \rightarrow \mathbb{R}_{\ge 0}$ is differentiable map, with $P(x) = 0 \ \forall x \in \mathcal{X}$ and $P(x) > 0 \ \forall x \in \mathcal{X}^c$. Further, suppose $P$ has ...
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### Monotone rearrangements of function, constrained optimization in $\mathcal{L}^p$

By $\mathcal{S}$ let us denote the set of such step functions $f:[0,1]\to [0,1]$ that additionally satisfy: \forall_{ x>\frac{1}{2}} \ \ \lambda\Big(f=x\Big) \ = \ x\cdot \Big[\lambda\Big(f=x\Big)...