Questions tagged [constraints]

In mathematics, a constraint is a condition of an optimization problem that the solution must satisfy.

Filter by
Sorted by
Tagged with
-1
votes
1answer
61 views

How can I linearize the IFF-THEN constraint with binary and continuous variable?

I have an optimization problem where $a_{m,n}\in\{0,1\}$ is a binary variable and $0\le f_{m,n}\le 1$ is a continuous vaiable I have an Iff-THEN constraint like this IFF $a_{m,n}=1$, THEN $f_{m,n}&...
2
votes
2answers
2k views

Model Predictive Control

I have a few confusions about Model Predictive Control (MPC). Since they are all minor questions related to the same category, I ask them under one topic. In an article, the cost function is defined ...
1
vote
3answers
303 views

Circle and Locus _ ONLY PEN AND PAPER ALLOWED.

Q) Let T be the line passing through the points P(–2, 7) and Q(2, –5). Let $F_{1}$ be the set of all pairs of circles $(S_{1}$, $S_{2}$) such that T is tangent to $S_{1}$ at P and tangent to $S_{2}$ ...
1
vote
1answer
514 views

Connection Between Orthogonal Projection onto the Unit Simplex and the Softmax Function

Referring to papers Softmax to Sparsemax and Efficient Projections onto the L1-Ball, what is the relationship between a euclidean projection onto the probability simplex and applying the Softmax ...
0
votes
1answer
108 views

How to handle equality constraints in this problem?

Here is the problem setup \begin{equation} \begin{array}{c} \text{min} \hspace{4mm} \mathbf{b}^{T}_{}\mathbf{A}^{}_{}\mathbf{b}^{}_{} \\ s.t \hspace{5mm} \mathbf{b} \in \mathbb{R}^{N} \\ \hspace{9mm}...
9
votes
1answer
913 views

Conic by three points and two tangent lines

With the exception of degenerate situations, a conic is uniquely determined by five points lying on it. Likewise, five lines tangent to a conic uniquely define that conic. With four points and one ...
5
votes
1answer
2k views

How to use Karush-Kuhn-Tucker (KKT) conditions in inequality constrained optimization

I am trying to understand how to use the Karush-Kuhn-Tucker conditions, similar as asked but not answered in this thread. Assume the target function is given by $f(x)$, where $x$ a vector. Let $g(x) ...
2
votes
1answer
228 views

How to Solve this Boolean Equations?

I have a Boolean Equations, described as below, $$\neg \mathbf{x} = \mathbf{M}\cdot \neg(\mathbf{M} \cdot \mathbf{x})$$ in which $\mathbf{M}$ is an $n\times n$ Boolean matrix, and $\mathbf{x}$ is an $...
6
votes
1answer
544 views

Integer linear programming constraint for maximum number of consecutive ones in a binary sequence

Consider an integer programming problem with binary decision variables $x_1,\ldots,x_n \in \{0,1\}$. Im trying to model the constraint that enforces the maximum number of consecutive ones in ...
4
votes
2answers
93 views

Find the unit vector within a subspace with the minimum norm projection onto another subspace

Let $W$ and $V$ be subspaces of $\mathbb{R}^n$ with dimensions $m$ and $p$ respectively. I want to find the unit vector in $W$ whose projection onto $V$ has the minimum Euclidean norm. From geometric ...
3
votes
0answers
769 views

From constrained to unconstrained maximization problem

I have the following constrained maximization problem $$ \max_{X_1,X_2,...,X_i,...,X_N} \sum_{i=1}^{N}X_i f_i(X_1,...,X_N) \hspace{0.2 cm} \text{subject to} \sum_{i=1}^{N}X_i-B\leq 0 \text{ and } X_i\...
3
votes
2answers
2k views

How to maximize area of two circles inside a rectangle without overlapping?

Two circles have to be drawn inside a rectangle of dimensions $W\times H$ such that the area of both circles is to be as large as possible without overlapping. Let the radii of the circles be $r_1$ ...
2
votes
1answer
64 views

How does one use the 'input/hr' column in the table below in setting up the problem?

I have to set up a linear programming problem corresponding to the following scenario: If my understanding of the problem is correct, I use $mod$: Let $i$ be $A$ or $B$. Let $x$ be amount of raw ...
2
votes
4answers
223 views

Binary variables in time series: integer linear programming

I'm working on a problem and I can't seem to find an easy solution to it. It's about an optimization problem, concerning a time series. I have a binary variable $\alpha_t$ for $t \in [0, 24[$. I ...
1
vote
2answers
92 views

Lagrange multipliers - confused about when the constraint set has boundary points that need to be considered

Consider the constraint $$S_1 = \{(x, y) \; |\; \sqrt{x} + \sqrt{y} = 1 \}$$ How to use Lagrange Multipliers, when the constraint surface has a boundary? In this case, after the Lagrange multiplier ...
1
vote
0answers
51 views

Help needed to define a constraint in an optimization problem?

Given objective function is \begin{align} \underset{\mathbf{p},\mathbf{q}}{\text{min}}\hspace{4mm} (\mathbf{p*q})^T \mathbf{A}(\mathbf{p*q}) \hspace{4mm} \\ s.t \hspace{4mm}\mathbf{p^Te_p}-1=0\\\...
1
vote
1answer
230 views

Optimal transport with relaxed constraint on marginals

Let $X$ be some appropriate space (metric measure, Polish, whatever...) and $X\times X$ the product space with $\pi^1$ and $\pi^2$ as projections onto the first and second factor, respectively. Let $\...
1
vote
1answer
107 views

Least squares minimization subject to the constraint $\sum_j|u_j| \leq 1$

I would like to learn about methods for minimizing the cost $L(u) = (f - Au)^{tr}M(f-Au)$, where $f \in R^m$ is a known vector, $u \in R^n$ is the argument of the minimization and $A \in R^{n\times m}$...
1
vote
2answers
399 views

$ {L}_{1} $ (L1) Norm Regularized Minimization with of Convex Function with Linear Equality Constraint Using ADMM Framework

In section 6.3 of this note there is a method for minimizing a loss function with l1 regularization. i.e. minimize $l(\bf{x})+\lambda||x||_1$ How can I add the equality constraint $\sum\limits_{i} ...
1
vote
2answers
2k views

Max and Min using Lagrange Multipliers

Suppose A is a symmetric matrix. Show that the maximum and minimum of $\mathbf x ^T A \mathbf x$ subject to the constraint $\mathbf x ^T \mathbf x=1$ are the maximum and minimum eigenvalues of A. I ...
0
votes
1answer
81 views

LP problem involving producing assemblies

I have to construct an LP problem based on the ff scenario that might be similar to a scenario in another question (in the sense that I felt the need to use $mod$): The productivities are minutes per ...
0
votes
0answers
313 views

How to Solve Boolean Matrix System?

I have a Boolean Matrix System (BMS) as described below $$Ax=c$$ where $A$ is a $n\times n$ Boolean matrix (i.e., all entries are either 0 or 1), $c$ and $x$ are two $n$-dimensional Boolean column ...
4
votes
1answer
239 views

Asymptotics of Gaussian integral over the unit sphere

I would like to evaluate the integral asymptotically over the unit sphere surface $$ Z =\int e^{a \cos^2 \theta + b \sin^2\theta\cos2\phi + c\cos\theta} d\Omega = \int\limits_{0}^{\pi}\int\limits_{0}...
3
votes
2answers
1k views

How to model a consecutive binary constraint?

Let us say we have $n$ binary variables $x_i$ for all $i=1,2,\ldots,n$, i.e., $x_i\in\{0,1\}$ for all $i=1,2,\ldots,n$. I need to write the following constraint: If $x_i=1$ and $x_{i+2}=1$, then $...
2
votes
1answer
242 views

Operations Research - Optimal Transport Routes

I have a problem in which there are 4 vessels available to transport people from 3 different bases back to a main base. Vessel 1 has a capacity of 50, can make 6 round trips and is allowed to visit ...
2
votes
2answers
406 views

Find the optimal solution without going through the ERO's

All I got is that $$12y_1 + 7y_2 + 10y_3 = 2(0) + 4(10.4) + 3(0) + 1(0.4)$$ and $y_2 = 0$ because $x_6$ is in basis. How do I find $y_1$ and $y_3$ without going through the simplex method? I took ...
2
votes
1answer
759 views

Why use Bordered Hessian than “simple” Hessian as second derivative test?

Why use Bordered Hessian than "simple" Hessian as second derivative test in a multi constrained optimization problem? The critical points are found from the Lagrangian so they follow the constraints. ...
1
vote
2answers
5k views

How do you find redundant constraints for a feasible region?

I've found a few papers that deal with removing redundant inequality constraints for linear programs, but I'm only trying to find the non-redundant constraints that define a feasible region (i.e. I ...
1
vote
0answers
44 views

Convexifying the biconvex constraint $xy-f(x,y)=0$ to $\max f(x,y)$

I want to solve $$\min_{x,y} \quad -f(x,y) \qquad \text{subject to} \quad xy C - f(x,y)K -\kappa = 0$$ where $x, y$ are from a compact and convex subset of $\mathbb{R}^2$ $f(x,y)$ is concave, in $...
0
votes
1answer
33 views

Write logical operator all(x<a) in terms of Heavyside functions

I am currently solving a complex optimisation problem, with constraints that take the form: $1 - all(g(x)<a) <= 0$, meaning I require all values $g(x)$ (for some function $g$) to be below some ...
0
votes
1answer
111 views

Is this convexification of non-convex constraint correct? [duplicate]

Original problem setup How to handle equality constraints in this problem? \begin{equation}\tag{1} \begin{array}{c} \min_{\mathbf{b}} \hspace{4mm} \mathbf{b}^{T}_{}\mathbf{A}^{}_{}\mathbf{b}^{}_{} \\...
0
votes
1answer
199 views

Minimize convex function with concave equality contraint

Let (1) $f:R^k \rightarrow R$ be a convex and differentiable function and (2) $g:R^K \rightarrow R$ be a concave and differentiable function. Consider the minimization problem $\{\min _x \ \ f(x) \ \ ...
0
votes
0answers
127 views

Compositional data constraints in optimization

Problem As part of an optimization problem I have a matrix $\Gamma$ that is the response in a multivariate regression problem $$\Gamma = \mathbf{AB}$$ where $\mathbf{A}$ is a matrix with $m \times ...
0
votes
1answer
39 views

How can we prove that we can get any $f$ modulo a prime $p$, that satisfies these equations?

We can suppose that we have four naturals not equal to zero: $a, b, c, d$. Further, we're working modulo a prime $p$. Now if we find $a, b, c, d$ that satisfy: $$f \equiv a \cdot c \equiv b \cdot d ...
-1
votes
1answer
45 views

$a$ is not equal to $b$ constraint and $a$ and $b$ have different domain. How to express it in predicate logic notation

This post is an extension of my previous post. Suppose I have an Excel (or a csv or whatever 'paper printed table') with 2 columns: $A$ and $B$. Example: ------- A | B ------- 1 2.5 7 5.5 8 ...
-1
votes
1answer
2k views

Linear Programming constraint equivalent of conditional

I would like to use the following conditional in my linear program: if(A == 1) then B = C + 1 A = binary, B and C are continuous. In the else case, any relation ...