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-1 votes
0 answers
41 views

Is it possible to backtrack an optimization solver? [closed]

I have an optimization problem and was using a linear programming optimizer to find solutions. However, I find that past a certain size, the problem becomes "infeasible" and has no solutions....
0 votes
0 answers
21 views

Easy instance of set cover

I am trying to prove that a natural greedy algorithm solves the following instance of the set cover problem: for a set of elements $e\in U$ with a set of weights $w_e$, we define the cost of a subset ...
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,\...
0 votes
1 answer
114 views

Mixed integer program and continuous Diophantine approximation

Let $n\in\mathbb{N}$ such that $n\geq 2$ and let $0<r<1$ be a real number. We wish to solve the following problem. $$\min_{(t,(z_j)_{j=2}^n) \in \mathbb{R}\times \mathbb{Z}^{n-1}} t$$ subject to ...
3 votes
0 answers
105 views

Techniques for solving linear inequalities

For $n$ real variables $x_1, \ldots, x_n$, I have a bunch of inequalities of form $2 x_i > x_j + x_k$ or $2 x_i < x_j + x_k$, where $i,j,k$ are distinct. My goal is to determine whether this set ...
0 votes
0 answers
39 views

Max-flow modeling with unified vehicle and commodity variables

I am working on a network flow problem that involves routing through a time-space network. The network consists of: A single source node and a single demand node. A fleet of vehicles with specified ...
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 $\mathcal{B}\subset \mathbb{R}^{k}$ (...
0 votes
0 answers
64 views

Alternatives to McCormick Envelope

I have an optimization problem for which I have the optimal solution obtained by the ILP. However, when I introduced the McCormick Envelope to replace the product of a bi-linear term in its LP ...
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 ...
1 vote
1 answer
181 views

Linear programming with "nice" matrices

Consider the following linear programming problem \begin{array}{ll} \text{minimize} & \mathrm 1^{\top} \mathrm x\\ \text{subject to} & v\le \mathrm A \mathrm x \le u\\ & \mathrm x \geq ...
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}^...
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{...
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 \...
1 vote
0 answers
96 views

On optimizing a multivariate quadratic function subject to certain conditions

The problem is to maximize $f(x_1,x_2,\cdots,x_n)=\sum\limits_{i=1}^{n}\Big(x_i-k_i\Big)^2$ for $n\ge 3$ subject to the conditions (1) $\sum\limits_{i=1}^{n}x_i=\sum\limits_{i=1}^{n}k_i\le n(n-1)$ ...
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, ...
0 votes
1 answer
405 views

Computing discrete optimal transport

I am trying to find a combinatorial approach to solve the following optimization problem. \begin{align} &\max_{x_{ij}} C_{ij} x_{ij}, \\ &\text{such that},\\ &\sum_{j} x_{ij} \leq r_i~\...
0 votes
1 answer
93 views

How quickly can this IQP or its MILP relaxation be solved

Let $A\in\{0,1\}^{(n,n)}$ be a $n$ by $n$ boolean matrix (in particular think of an adjacency matrix of a graph), and consider the following optimization problem: $$\begin{align*}&&\max_{P\in\{...
0 votes
1 answer
540 views

Method for (binary) optimization under constraints

I would like to know if there is a method to solve the Problem. Problem: Maximize the following function: $$f(p_{1,i},p_{2,i},\dotsc,p_{m,i})=\sum_{i=1}^{n}\begin{bmatrix}p_{1,i} & p_{2,i} & \...
1 vote
0 answers
59 views

How do I incorporate Ito's lemma into the solution for a finite-horizon stochastic cake-eating problem?

I'm interested in finite-horizon, continuous-time cake-eating problems in which the agent has a time-horizon $W$ over which to eat the cake, and then chooses an optimal consumption path $\{h_t\}_0^W$, ...
2 votes
1 answer
227 views

Solving linear programming without solving linear programming

Let $v_1, \cdots, v_n$ be vectors in $\mathbb R^k$, and let $M$ be the Gram matrix of them. It's possible to determine from $M$ and $k$ whether the only vector that has nonnegative inner product with ...
2 votes
1 answer
372 views

Who called Farkas' fundamental theorem a lemma?

Farkas proved his famous result (which, nowadays, is fundamental in optimization theory) in 1902 and called it Grundsatz der einfachen Ungleichung which may be translated as fundamental theorem of ...
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{...
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 $\...
1 vote
0 answers
35 views

How to chose the start vector for the MTZ variables

In the context of LP-formulations for the Traveling Salesman Problem the MTZ constraints prevent subtours via $n$ (i.e. effectively $n-1$) additional variables $$u_1=1\\2\le u_2,\,\dots ,\,u_n\le n\\ ...
0 votes
0 answers
96 views

Why is Gaussian distribution always chosen for smoothed analysis?

I came across the algorithmic perfomance analysis model of smoothed analysis. In all references that I read a Gaussian distribution was used for perturbation (e.g. Spielman and Teng 2004 for the ...
0 votes
0 answers
165 views

Minimum circumscribed ellipsoid of $\mathcal H$-polytope

Given matrix $A \in \mathbb{R}^{m \times n}$ and vector $b \in \mathbb{R}^n$, consider the $\mathcal H$-polytope $P$ defined as follows $$ P := \left\{ x \in \mathbb{R}^n : Ax \leq b \right\} $$ I ...
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 ...
0 votes
1 answer
116 views

Iterations of Dantzig-Wolfe Decomposition for a Simple Linear Programming problem

This arises from an engineering problem I am working on. Let $\mathbf{c}_i,\mathbf{a}_i,\mathbf{b}_i\in \mathbb{R}^{d}$ be a given set (collection) of vectors where $i\in\{1,\dots,n\}$. Define the ...
1 vote
1 answer
98 views

Optimality gap between a joint linear program and decoupled sub programs

Let $\mathbf{c}_i,\mathbf{s}_i$ be given entry-wise positive $n\times 1$ vectors for $i\in[1,\dots,d]$. Let $\tau, \alpha_1,\dots, \alpha_d$ be given positive constants. Consider the linear ...
1 vote
0 answers
81 views

Algorithm for deciding feasibility of linear programs [closed]

Suppose I have the simple linear program $$Ax \geq 0, \quad x \geq 0$$ We know that this system has a solution (for example, $x=0$). But, what if we made this rule for this system? $$Ax \geq 0, \quad ...
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: ...
0 votes
1 answer
76 views

A question on graph partitioning

Given a connected un-directed simple graph $G=(V,E)$, is there a polynomial time algorithm to find the smallest subset $S$ of $V$ such that each node in $V \setminus S$ has at least 50% of its ...
2 votes
1 answer
139 views

linear programming with $n$ choose $r$ variables

Given parameters $r < n$, define $m = {n \choose r}$ and let $A$ be the $n\times m$ matrix whose columns are all the vectors with $r$ $1$'s and $n-r$ $0$'s. Let $b$ be a positive $n$-vector. Is ...
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 $b$ ...
2 votes
1 answer
763 views

Integer solution of optimal transport

Let us consider two vectors $\mathbf{a}=(a_1,...,a_n)$ and $\mathbf{b}=(b_1,...,b_m)$ so that each quantity is an integer $a_i,b_j \in \mathbb{N}$. It represents for example supply and demand. Let $\...
1 vote
1 answer
3k views

How to minimize l1-norm constrained by "infinity norm"

Let $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m $. I have the following two problems: P.1. \begin{equation} \underset{x\in\mathbb{R}^n}{\text{minimize}} \| Ax-b \|_1 \\ \text{s.t. } \| x \...
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+\...
1 vote
0 answers
148 views

Convergence of infinite linear programming

Suppose we have the following linear program (LP1), $$\min_{f \in \mathcal{C}} \int_{\mathbb{R}} f(t) \cos(2 \pi x_0 t) dt \\ \text{subject to } \int_{\mathbb{R}}f(t)dt = 1 \\\forall x\in [0,1]: \int_{...
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 ...
5 votes
3 answers
1k views

Algorithm for the intersection of a vector subspace with a cone of non-negative vectors

Hi, I would like to know whether there is some more effective way of how to compute an intersection of a vector subspace of $\mathbb{R}^{n}$ with a cone of vectors with non-negative entries than the ...
0 votes
2 answers
4k views

Linear programming piecewise linear objective

I am fairly new at linear programming/optimization and am currently working on implementing a linear program that is stated like this: max $\sum_{i=1}^{k}{p(\vec \alpha \cdot \vec c_i)}$ $s.t. $ $|\...
1 vote
0 answers
163 views

Can we reduce the maximization of this integral to the maximization of the integrand?

I would like to know whether we are able to reduce the following optimization problem to the pointwise optimization of the integrand (or how we can solve it otherwise): Maximize $$\sum_{i\in I}\sum_{j\...
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 ...
0 votes
0 answers
101 views

How can we analytically solve this max-sum-min problem?

Let $I$ be a finite set, and $A_{ij},B_{ij},x_i,y_j\ge0$. I want to find the choice of $x_i,y_j$ maximizing $$\sum_{i\in I}\sum_{j\in J}A_{ij}\min\left(x_i,B_{ij}y_j\right)\tag1$$ subject to $$\sum_{i\...
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....
3 votes
2 answers
331 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 ...
1 vote
0 answers
24 views

Simple monotonicity property for coordinate descent and linear objective functions

Let $S \subset \mathbb{R}^n$ satisfy $0\leq x_1\leq\dots\leq x_n$ for all $\mathbf{x}\in S$, among other (possibly nonconvex) constraints, and suppose in addition that $\sum_{i=1}^n x_i \geq 1$ for ...
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 $>1$) $\textbf{The problem at hand is as follows:}$ ...
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}\|...