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-1
votes
2answers
81 views

Is finding a local minimizer of a NP-hard optimization problem is still NP-hard [closed]

I was wondering if for a NP-hard optimization problem, I only want to find its local minimizer, is it still NP-hard or NP-hard is only true when trying to find a global minimizer?
1
vote
0answers
12 views

Deterministic global solution to find the Optimal-knot placements for continuous piecewise linear functions to fit nonlinear data

I have been searching lately for a deterministic global technique to linearize a nonlinear function with continuous piecewise linear regions. I've a univariate nonlinear function y=f(x). where f(x) ...
0
votes
0answers
23 views

solution of an infinite horizon optimization problem

Give the following formulation: $\min_{\{x_s(t):\forall s,t\}} \sum_{s \in \mathcal{S}} \mathbf{1}\left(\lim_{T\rightarrow \infty} \frac{1}{T} \sum_{t=1}^T \frac{y_s(t)}{x_s(t)}\leq 1\right)$ $s.t. ...
2
votes
0answers
85 views

Solve non-linear Optimization Problem [closed]

I have to find $x$ that minimizes: $$ \sum_{k}(x^H\textbf A_kx - b_k)^2$$ where $A_k$ are 4 x 4 positive definite matrices ($A_1, A_2,...A_k$), $x$ is 4 x 1 vector and $b_k$ are scalars ...
1
vote
1answer
88 views

Convert general optimization problem to LP problem

I am trying to convert the following problem into a linear programming problem: There are $M\times N$ matrix $T$ of real numbers between 0 and 1 and $N\times 1$ vector $w$ of real numbers between 0 ...
5
votes
1answer
52 views

Two fold optimization: is there an established approach for this kind of problem?

I have a list of thousands of linear expressions, with less than 100 total variables. Each expression is associated with a positive point value. I want to make as many of the expressions greater than ...
0
votes
0answers
76 views

Optimization over space of probability measures

Consider an optimization problem as follows: $$ \min\mathbb E_w[f_0(w)] \mathrm{\,\,\,\,\,\ s.t.\,\,\,\,} E_w[f_i(w)]\leq 0 ,\,\,\, i=1,\dots, k $$ where the maximum is taken over $\mathscr M$, ...
4
votes
0answers
139 views

maximize non-convex composite function

I want to maximize a composite function over a convex set \begin{equation} \begin{aligned} & \underset{\mathbf{p}}{\text{maximize}} & & f(\mathbf{p})-g(\mathbf{p})\\ & \text{subject ...
4
votes
0answers
67 views

Basin of Attraction

I have a function $F$ which is defined as follows: $$ F(x) = \sum_{i=1}^N f(z_i^T x) $$ where ${z_i}$ are known $m \times 1$ vectors, $x$ is an $m \times 1$ vector, and for $t\in \mathbb{R}$, $f(t) = ...
0
votes
1answer
84 views

Why does optimization of a sum of two terms result in “neat” answers? [closed]

This is a somewhat vague and philosophical question. Consider the following three problems: Problem 1: Minimize over all real-valued $x,$ the function $f(x) = bx-ax^2$ where $a,b>0.$ ...
2
votes
1answer
49 views

Bounding the difference in the value of a strongly convex function at its integer minimum and other integer points

I am currently working on a problem where I have to minimize a $m$-strongly convex function $$f ~: ~\mathbb{R}^n \rightarrow \mathbb{R}^+$$ over a bounded integer lattice, $$L = \mathbb{Z}^n \cap ...
9
votes
3answers
355 views

“Most Similar Vector Problem” on an Integer Lattice?

I am currently working on problem that I think could be expressed as an integer lattice problem. Given $u \in \mathbb{R}^n$ and a bounded integer lattice $L = \mathbb{Z}^n \cap [-M,M]^n$ I would like ...
2
votes
2answers
184 views

Maximal minimum for a sum of two (or more) cosines

Please prove (or disprove, and give the correct answer): $$2 =\mathrm{argmax}_{r\geq 1}\min_{x\in \mathbb{R}}\left[\cos\left(x\right)+\cos\left(rx\right)\right] $$ In other words, find $r \geq 1$, ...
1
vote
1answer
70 views

arg min_X ||A X B - C||^2, with X diagonal [closed]

Let $A, B, C$ be known matrices, and let $X$ be an unknown matrix. Given that $C = AXB \Leftrightarrow \text{vec}(C) = K \text{vec}(X)$, where $\text{vec}(\cdot)$ denotes the vectorization of a ...
0
votes
1answer
217 views

Maximize a sum of log of sum

For a matrix $c (m\times n)$ of non-negative constants, find values of $\lambda_1, \lambda_2, \ldots, \lambda_n$ that satisfy $\sum_{k=1}^n \lambda_k = 1$, $\lambda_k \ge 0 \, \forall k$ and maximize ...
0
votes
0answers
42 views

Global optimisation of the real part of impedance

I have the following global optimisation problem: $$ \underset{\omega}{\min}-c^{T}\left(\omega^{2}\mathbf{1}+A^{2}\right)^{-1}b $$ where $A$ is a $n \times n$ real matrix, $c$ and $d$ are ...
2
votes
1answer
143 views

Is there any algorithm can find local minima of nonconvex objective function in guaranteed polynomial time?

More precisely, The setting could be formulated as, $min. F_{\lambda}(p)$ over permutation matrices $P$ Here $F_{\lambda}(p)$=$\lambda *F_{0}(p)+(1-\lambda)F_{1}(p)$ where both $F_{0}(p)$ and ...
1
vote
0answers
85 views

Likelihood convexification

I am doing constrained vector optimization using a non-convex non-linear likelihood function. My problem is of the following form: $$\begin{align*}\hat Q &= \underset{\vec Q}{\arg\min} -\log ...
6
votes
0answers
110 views

A specific case of the $p$-center problem

Given a fixed positive integer $m$, let $\cal{S}$ be the subset from $\mathbb{R}^m$ defined as $\cal{S} = \{u \in \mathbb{R}^m \mid \forall i \in \{1, \dots, m\}, u(i) > 0$ and $\sum_{i=1}^m{u(i) = ...
2
votes
0answers
170 views

Solution of a linearly constrained quadratic programming problem [closed]

What is the solution of the following optimization problem: \begin{align} &\min{\mathbf{p}^\mathrm{T} \mathbf{B} \mathbf{p}}\\ &\text{subject to}: \mathbf{0}\leq{\mathbf{p}}\leq \mathbf{1}. ...
0
votes
2answers
67 views

Analytic solution $\underset{n} {\mathrm{argmin}} \frac{a}{r + ns} + \sum_{i=0}^{n-1}\frac{b}{r + is}$ [closed]

Could anyone provide some hints for solving: $\underset{n} {\mathrm{argmin}} \frac{a}{r + ns} + \sum_{i=0}^{n-1}\frac{b}{r + is}$ for $n \in \{1,2,3,\ldots\}$ The problem is part of a coding ...
1
vote
0answers
46 views

Modified Orthonormal Procrustes Problem

In the general orthonormal Procrustes problem, we want to find an orthonormal matrix $C$ to minimize $\|Y-XC\|_F^2$, where $Y$ is a known $n\times q$ matrix, $X$ is a known $n \times m$ matrix, and ...
1
vote
2answers
163 views

iterative solution better than analytic solution? [closed]

My supervisor and I were discussing a specific optimisation problem this afternoon. To be simple: solve for $R$ in the equation $Rx=y$, where $x$, $y$ are made of samples in two difference ...
2
votes
1answer
169 views

Maximum of a mollified/convolution function

I have a function $f:{\mathbb R}\rightarrow {\mathbb R}_+$ which has a unique maximum at $x=0$. $f$ can be symmetric or asymmetric. I am interested on the mollified-f function ...
0
votes
0answers
71 views

numerical solver for stochastic optimal control problems

can any one recommend numerical solver (c/c++ library preferred) for stochastic optimal control problems? For deterministic optimal control I found something like that: ...
1
vote
1answer
150 views

maximizing a function involving factorial

Can someone suggest a way to calculate the maximum with respect to $x \ge 1$ of: $$f(x)=\frac{1}{x!} \frac{1}{1-c^{1/\binom{x+n-1}{n-1}}}.$$ The constants $c$ and $n$ are parameters such that $c \in ...
2
votes
2answers
838 views

How to fit the parameters of differential equations with known data?

I have the following data from chemical kinetics research to fit the parameters of ordinary differential equations: $$ \left[ \begin{array}{ccccccc} \text{No.}& t & y_1(t)&y_2(t) & ...
1
vote
0answers
42 views

Changing a nonlinear equality constraint into some conic inequality plus rank constraint

If we have a constraint optimization problem in which one of our constraint is $\prod\limits_{k = 1}^N {\left( {x - {a_k}} \right) = 0} $ . How could this nonlinear equality condition be changed into ...
9
votes
4answers
295 views

Software tools for medium-scale systems of polynomial equations

I am attempting to find all real solutions of a system of 12 polynomial equations in 12 unknowns. The equations each have total degree 6 and contain up to 1700 terms. I am only interested in real ...
0
votes
2answers
189 views

Uniqueness of solution of a nonconvex optimization problem

What conditions need to be hold for a nonconvex optimization problem to have a unique solution? Specifically, I have the following minimization problem that I'd like to know whether it has a unique ...
1
vote
1answer
311 views

Convergence rate of stochastic gradient decent with projections

Given a strong (not only strict) convex function $f: \mathbb{R}^n\to\mathbb{R}$. On such problems, stochastic gradient decent (SGD) has a convergence rate of $O(1/T)$, where $T$ is the number of ...
0
votes
0answers
66 views

Sufficient optimality condition for a non-smooth quasiconvex problem

The result of relaxing to an integer program is the following optimization problem: $$\min_{\textbf{x}} \sum_{i=1}^n \alpha_i h(x_i)\quad subject \; to \quad A\textbf{x} = \textbf{0}$$ where ...
1
vote
0answers
100 views

range of the difference-of-two-qubit-$4 \times 4$-density-matrix-determinants

The determinant of a two-qubit $4 \times 4$ density matrix--that is, a Hermitian, nonnegative definite matrix with unit trace--lies between $0$ and $(\frac{1}{2})^8$. (A "pure state" has determinant ...
2
votes
2answers
280 views

Finding the maximum of a multivariate polynomial of degree one

I need to find the global maximum of the function \begin{align} f\left(x\right) & = p_1 \max\left(\sum a_{1i} x_{1i}, \sum b_{1i} x_{1i}\right) - \sum c_{1i} x_{1i} \\ &+\ldots \\ &+ p_n ...
1
vote
0answers
123 views

Maximizing an integral over a convex region

Let $C$ denote a compact, convex region in the plane containing the origin with unit area, and let $f$ be a probability distribution on $C$. Let $f^\ast$ denote the distribution that maximizes the ...
2
votes
1answer
119 views

A Recursive Maximization Problem

Let $A\ge B>0$ be real constants. I say that a function $f:[0,1]\rightarrow[0,1]$ satisfies the $(A,B)$-condition if for all $p\in [0,1]$, the expression $$q(A-Bp-Bf(q))$$ is maximized (not ...
1
vote
0answers
157 views

Constructing an $\epsilon$-net for a Lipschitz subspace of $L^2$

Let $X$ be a subset of $L^2([0,1])$ which contains only Lipschitz function. Also, the member of $X$ are uniformly bounded $$ |x(t)| < M, \text{ for all $x \in X$ and $t \in [0,1]$}. $$ Let $F: X ...
23
votes
4answers
1k views

Why are optimization problems called “programming”?

Why are optimization problems often called programs? linear programming geometric programming convex programming Integer programming ...
3
votes
1answer
103 views

Compactness of a semi algebraic set

Suppose I have a polynomial $p\in R[x_1,\ldots,x_n]$ and I look at the set $S:=\{ x\in R^n : p(x)\geq 0\}$. Are there algebraic certificates on $p$ that will certify that $S$ is compact?
1
vote
1answer
176 views

Maximum Dispersion of a Connected Geometric Graph

Let $\left\{\mathbf{p}_1,\dots, \mathbf{p}_k\right\}$ be a set of points in $n$-dimensional Euclidean space, and let the second moment of these points be defined as: $ U=\sum \limits_{i=1}^{k} ...
3
votes
1answer
245 views

Optimization problem - maximizing number of satisfied linear inequalities subject to a quadratic constraint

I am wondering what is known about optimization problems of the following type. Our control x is a unit vector in $\mathbb{R}^n$. We are given a finite number of linear inequalities $$Az≥b,$$ and we ...
3
votes
2answers
235 views

Uniqueness of fixed points for rational transformations

Background Let $a,b,c,d$ be nonnegative constants and consider the map $T\colon [0,1]\times[0,1] \rightarrow [0,1]\times[0,1]$ defined by $$ T(x,y) := \left( \frac{1}{1 + ax + by}, \frac{1}{1 + cx + ...
2
votes
1answer
654 views

Maximizing supermodular functions

I have a real supermodular objective function which I want to maximize with constraint. The constraint is on the size, like |A|=k . I am wondering if anyone can give me more information about a ...
0
votes
0answers
107 views

The role of subgradient in programming with nonsmooth functions

It is obvious that there is similarity between subgradient and gradient. The subgradient of smooth functions is reduced to gradient. I have two questions. The first is does there exist subgradient ...
2
votes
1answer
506 views

lipschitz constant of a multivariate function

I have a function $f:\mathbb{R}^{50} \rightarrow \mathbb{R}$ and I need to compute the Lipschitz constant of $f$ to solve an optimization problem using a specific algorithm. Does any one have ...
0
votes
1answer
96 views

Avoiding epsilon in mixed integer linear and quadratically constrained programs

I would like to represent the following constraint as MILP constraint where $x \in [a, b]$ with fixed $a, b \in \mathbb{R}$ and $y \in \lbrace 0, 1 \rbrace$. $(x = 0 \wedge y = 1) \vee (x \neq 0 ...
6
votes
1answer
256 views

Generalization of the equilateral triangle?

I consider points in the two-dimensional plane. An equilateral triangle is a set of three points in the plane which are equidistant. Suppose now I have $n$ points $x_1,...,x_n$. What is the ...
1
vote
1answer
352 views

for what arguments the function reaches maximum?

Hi, What is the maximum of the following function?: $f(x_i,w_i)=\frac { \sum w_i}{ \sum \frac {w_i}{x_i} } - \frac{ 1 - \prod \left ( 1 - w_{i}\right )}{ 1 - \prod \left ( 1 - \frac{w_{i}}{ ...
1
vote
1answer
141 views

What kind is this optimization problem

I come across a problem like $\max {\frac{1+v}{1-u}}$ $s.t.~$ $ux^2+vy^2-xy\ge0$ $\forall x,y\in\mathbb{R}$ I do not know much of optimization. What I have done is that $ux^2+vy^2\ge ...
2
votes
1answer
71 views

Computing a point of refraction

Oddball question: say I want to travel from $(a, b)$ where $b > 0$ to $(c, d)$ where $d < 0$ using the shortest path, where I can travel at velocity $v_1$ in the upper half-plane and at velocity ...