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2 votes
0 answers
119 views

Seeking insights on bounded set positive solutions for a set of linear systems in $\mathbb{R}^n$

Before delving into my query, I'd like to provide some context. Consider a continuous function $f:\mathbb{R}^{k}\rightarrow\mathbb{R}^{m}$ and a compact set $\mathcal{B}\subset \mathbb{R}^{k}$ (...
Diego Fonseca's user avatar
1 vote
0 answers
94 views

Linear Program Optimal Value

If $f(A,b,c)$ is the optimal value of a linear program $\min c.x$ subject to $A.x \leq b ; x \geq 0.$ Does $f(A,b,c)$ have a piecewise polynomial/rational upper bound in $(A,b,c)$ on the domain of ...
Pathikrit Basu's user avatar
0 votes
0 answers
55 views

Relationship of optimal solutions between the total function and the sub function

This is an unconstrained convex optimization problem. Let $\mathcal{N}=\left\{1,\ldots,n\right\}$, $2\leq n<\infty$. Suppose there are many strongly convex functions $f_i(x)$, where $x\in\mathbb{R}^...
lzzz's user avatar
  • 1
0 votes
1 answer
103 views

Constrained linear optimization problem on $C^1$

I am dealing with a problem of the form ($a<b$) $$ \displaystyle \max_{v \in C^1([a, b])} \int_a^b v(x)~\mathrm{d}x, \quad \mathrm{s.t.} \int^b_a \big(-o'(x)v(x)-v'(x)o(x)\big)f(x)~\mathrm{d}x \...
Hyperbolic PDE friend's user avatar
0 votes
1 answer
147 views

Is there a redundant constraint in linear programming? [closed]

From wikipedia: But... Why do we need the $x\ge 0$ part? We can instead do $-x\le 0$, and thus saving a line in the definition (which is not a big deal but nevertheless nice). (In order to do that, ...
Bipolo's user avatar
  • 3
1 vote
0 answers
98 views

Solution of a simple optimization problem

Let $\mathbf{U}_1$ and $\mathbf{U}_2$ be two arbitrary unitary matrices and $\mathbf{D}$ be a diagonal matrix. What is the solution of the following optimization problem? \begin{align} \min_{\mathbf{...
Math_Y's user avatar
  • 287
0 votes
0 answers
124 views

The best unitary matrices that approximate a matrix product

Let $\mathbf{A}$ be an arbitrary $N\times N$ complex matrix. Moreover, $\mathcal{U}_1$ and $\mathcal{U}_2$ are distinct subsets of all unitary matrices. Suppose the matrices $\mathbf{U}_1$ and $\...
Math_Y's user avatar
  • 287
0 votes
0 answers
137 views

Any technique for linearization, or linear approximation?

Consider the following Matrix constraint: $$ \begin{bmatrix} -U+\psi\Sigma_b^{-1} & V \\ V^T & -V^TU^{-1}V+\tau_2 -\psi \end{bmatrix} \leq 0 $$ where $\Sigma_b$ is a known positive definite ...
Navid Hashemi's user avatar
1 vote
0 answers
920 views

Maximizing a piecewise-linear convex function

Crossposted on Operations Research SE. I am working on an optimization problem where some of the terms of the objective function to maximize are expressed as a piecewise linear function of variables: ...
lovasoa's user avatar
  • 111
3 votes
0 answers
122 views

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+\...
Math_Y's user avatar
  • 287
2 votes
0 answers
46 views

Notion of distance between linear programs

Consider the linear programming problem \begin{align} \max_{x}&~c^Tx \\~s.t.~~a^Tx &\leq B~,~0\leq x_i \le1 \end{align} where $c$ and $a$ are $n \times 1$ given non-negative vectors. $B$ is a ...
dineshdileep's user avatar
  • 1,421
0 votes
1 answer
320 views

Sub optimal algorithm for linear programming

Consider the linear programming problem \begin{align} f^* = \max_{x}&~p^Tx~\\~&A^Tx\leq b~,~0\leq x_i\leq 1 \end{align}where $c$ is a $n\times 1 $ vector, $A$ is a $n\times c$ matrix and $b$ ...
dineshdileep's user avatar
  • 1,421
0 votes
1 answer
113 views

How do I solve this integer programming problem with non convex constraints?

I am not sure if this is the right place to post this question, please point me to the correct forum if I posted in a wrong place. I have an optimization problem like this ...
Aaron_Geng's user avatar
0 votes
1 answer
61 views

Variant of the linear programming problem

Good afternoon, my experience in mathematical programming is low. I would like to know if there is any general method to address the following problem: $$\text{Minimize }\sum_{i=1}^n d_i(x_j)$$ $$s.a....
Rusbert's user avatar
  • 193
13 votes
3 answers
835 views

Famous theorems that are special cases of linear programming (or convex) duality

The max flow-min cut theorem is one of the most famous theorems of discrete optimization, although it is very straightforward to prove using duality theory from linear programming. Are there any ...
0 votes
1 answer
145 views

How to solve this optimization problem efficiently? [closed]

Let, $D\in\mathbb{C}^{1\times M}$ is a row vector with $M$ elements $V\in\mathbb{C}^{3^M\times M}$ is a given matrix $T$ is a scalar (real and $>1$) $\textbf{The problem at hand is as follows:}$ ...
dipak narayanan's user avatar
1 vote
1 answer
123 views

LASSO problem but with a maximization instead of minimization

I have the following optimization problem (like the LASSO problem but with maximization instead of minimization): $\mathbf{maximize}_{\boldsymbol{\alpha}} \|\mathbf{x} -\mathbf{A}\boldsymbol{\alpha}\|...
Christina's user avatar
  • 111
4 votes
0 answers
539 views

Using Linear Programming as an iterative procedure

Suppose, we have a linear program and an optimal solution to it. Suppose now, we get a new constraint. We want to obtain an optimal solution to the given linear program extended by that new constraint....
D. Rusin's user avatar
  • 391
1 vote
0 answers
81 views

Maximizing sum of homogeneous functions of order one over a polytope

Let $f_i: \mathbb{R}^n\rightarrow \mathbb{R}$ be concave, increasing (i.e., if $x\geq y$ where the inequality is entry wise, we have $f_i(x)\geq f_i(y)$), and a homogeneous function of order one for ...
Ozzy's user avatar
  • 393
2 votes
0 answers
105 views

Optimization over a convex cone generated by a set is equal to optimization over the set

Within my research I found an important doubt and that prevents me from advancing, the context of my doubt is as follows: We considerer the following optimization problem $$ \left\{\begin{array}{cl} \...
matematicaActiva's user avatar
3 votes
1 answer
368 views

Lot sizing problem: how to add these cuts efficiently

Consider the set of constraints of the uncapacitated lot sizing problem: $$ \{(x,s,y)\in \mathbb{R}^n_+ \times \mathbb{R}^n_+ \times \mathbb{B}^n \;|\;s_{t-1}+x_t = d_t+s_t,\; x_t \le My_t,\; t=1,\...
Kuifje's user avatar
  • 225
0 votes
2 answers
120 views

Reference request: dependence on linear constraints

Excuse me if my question is stupid. I'm seeking the references on the dependence of the (linear) optimization problem on (linear) constraints. Namely, consdier the following optimization problem: $$P(...
CodeGolf's user avatar
  • 1,835
3 votes
0 answers
71 views

Dependence of optimization problem on the linear constraints

Let $I=\{x_1,\cdots, x_n\}\subset \mathbb R$ be fixed. Given two probability distributions $\alpha=(\alpha_i)_{1\le i\le n}$ and $\beta=(\beta_i)_{1\le i\le n}$ on $I$, and a matrix $c=(c_{i,j})_{1\le ...
CodeGolf's user avatar
  • 1,835
1 vote
0 answers
1k views

Analytic formula for minimizing the maximum inner product of a set of vectors

Given $x_j\in\mathbb{R}^n$, $j=1,\ldots,p$, find $$ \widehat{w} \in \arg\min_{\Vert w\Vert=1}\max_{1\le j\le p} |\langle w,x_j\rangle|. $$ I am also interested in the special case where we further ...
JohnA's user avatar
  • 710
2 votes
0 answers
64 views

Finding orthogonal basis with constraint

Is there any fast algorithm that output an orthogonal basis $e_i,i\leq n$ of $R^n$ with $e_i\in V_i$? Where $V_i,i\leq n$ are given linear subspaces of $R^n$. And is there any condition on $V_i,i\leq ...
Jiayi Liu's user avatar
  • 909
0 votes
1 answer
204 views

Is the linear production game a convex game?

In cooperative game theory, the linear production game (LPG) is defined by letting the characteristic function have the form of a linear programming problem. Does anyone know if the LPG is a convex ...
philipvn's user avatar
6 votes
0 answers
97 views

Finding the optimal mixture of two convex functions

I am trying to find an efficient way to solve the problem $$\min_{p,x_1,x_2} p\cdot f(x_1)+ (1-p) \cdot f(x_2)~~~~~ s.t.\\p\cdot g_1(x_1) + (1-p)\cdot g_2(x_2)\leq 1 \\ 0\leq p \leq 1$$ where $x_1,x_2\...
Robert Lowell's user avatar
4 votes
3 answers
1k views

Minimax theorem on a non convex domain

A minimax theorem is a theorem which states that under certain conditions on $\mathcal{X}$, $\mathcal{Y}$ and $f$: $$ \inf_{x \in \mathcal{X}}{\sup_{y \in \mathcal{Y}}{f(x,y)}} = \sup_{y \in \mathcal{...
Adrien's user avatar
  • 591