Questions tagged [constraints]

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

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Sum of functions at the minima and maxima with constraint

The Problem I have a sum of $N$ known functions $f_k(x)$, $$ y(x) = \sum_{0<k\leq N} k^2f_k(x) \tag{1}\label{eq1} $$ I also have a constraint which is valid at the minima and maxima points $x_m$ $$ ...
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Reformulate non-convex quadratic constraint to convex constraint?

Consider a simplified problem with two real variables $x_1$ and $x_2$, for which I I want to minimize $x_1^2 +x_2^2$ under the following quadratic constraints: $ d_1^2 \leq (x_1+x_2)^2 \leq d_2^2$ Is ...
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Calculus of Variation with inequality constraints - Lagrange multiplier approach does not work

This is related to a question I asked several years ago, see here. I want to find the functon $y$ which maximizes the functional $$J[y]= \int_{a}^{a+1} (2x- 1-a)y(x) \left(1+B\left[y(x)-1\right]\right)...
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Absolute max and min with lambda

The function $f(x,y)=xy$ has an absolute maximum value and an absolute minimum value subject to the constraint $x^2+y^2-xy=9$. I know $grad(f(x,y))=(y,x)$ and $grad(g(x,y) = (2x-y,2y-x)$ So how do I ...
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Constrained optimization and level curves

I'm currently working on this Optimization problem: $\min \max (|x-2|,|y+1|)$ Subject to $x,y\geq0$ We have been asked to show the optimal solutions graphically using the fact: $\max (|x-2|,|y+1|) = ||...
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Minimizing using simplex

I've been asked to minimize: $3x_1 -x_2 +2x_3$ using simplex. The conditions are: $x_1 +x_3 \geq 7$ $x_2-x_1\leq 5$ $x_2-2x_3 \leq 8$ $x_1,x_2,x_3\geq 0$ So I started by adding in my slack and '...
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Optimization with positive definite constraints

Suppose I have a function $f$ that is strictly convex on the positive semidefinite sets. I wonder if it is generally possible to minimize $f$ with positive definite constraints? $$\text{minimize } f(X)...
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InverseFunction[] with ArcTanh and Log [closed]

I solved an equation and one solution was this: $\text{InverseFunction}$$\left[2 A \log \left(A \sqrt{\text{$\#$1}^2 \left(A^2\right)-4 F}+\text{$\#$1} A^2+4 \text{$\#$1} B\right)-2 A \tanh ^{-1}\left(...
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How to constrain second order cone program so vector elements are either x or zero?

I have a $n$-length vector $v_i$ and I have the constraint $||v_i||_2 \leq \mu_i \forall i$ in my second order cone program (where $||.||_2$ is the L-2 norm). I want to add a constraint (either an ...
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Verify that $x = −(I − A^{\dagger}A)c$ optimises $ \frac{1}{2}\|x\|^2 + \langle c,x \rangle$ under constraints

Under the constraint that $Ax = 0$ and $A$ is full row rank, how would you get the solution to the optimisation problem $$\text{min } \frac{1}{2}\|x\|^2 + \langle c,x \rangle$$ My text says that $x = −...
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How the KKT conditions change when we replace the equality constraint $g(x)=0$ to $||g(x)||^2=0$?

Consider the following optimization problem: $$ \min_{x \in \mathbb{R}^n} f(x) \text{ subject to } g(x)=0 , $$ where $f$ and $g$ are smooth functions from $\mathbb{R}^n$ to $\mathbb{R}$. This is ...
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Optimization on manifold with additional linear constraints

I am looking for references about manifold optimization when additional constraints on the variable are present. Specifically, the problem I'm interested in is something along this line \begin{align} \...
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Extremising $\int_0^1 f(x) f(1-x) \ \mathrm{d}x$ subject to length of $f$ and endpoints

I have recently learnt some Calculus of Variations and was trying to apply this to a question I made: Over all functions $f: [0, 1] \to \mathbb{R}$ satisfying $f(0) = f(1) = 0$ with fixed curve length ...
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Second-order sufficiency conditions with trivial critical cone

I would like to use the projected Hessian to classify stationary points as saddle points or local minima via SOSC (or inconclusive). To better understand how to operationalize this task for numeric ...
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vehicle routing optimization, Big M method of reformulation of constraints

Please excuse me for the long question, if I dont prrovide this info. my post gets removed! The following optimization problem is called Mixed-Integer Quadratically Constrained Programming (MIQCP) ...
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What is the computational complexity (in flops) of nonnegative least square optimization?

Suppose I have a vector $x\in \mathbb{R}^D$ and a matrix $U\in\mathbb{R}^{m\times D}$. I would like to solve the following nonnegative least squares optimization problem: $a = \text{argmin}_{y\in \...
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In the Constrained Optimization example provided in the post, how do we "not lose generality" and what does it mean?

I'm currently working through some examples and questions for some basic first-order constrained optimization problems. My current issue is how "loss of generality" works in this context: ...
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Condition within a condition in Constraint [closed]

How could I model my constraints: when $q=0\implies y=0$ and when $q=1$, \begin{equation} y = \begin{cases} % Case 1: d & \mathrm{if }\; d\leq b\...
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What is the standard name for the subspace of a vector space whose elements are orthogonal to the nullspace?

This seems easy, but I can't find a standard name. Suppose I have a vector space defined a set M of linearly independent columns and N rows (let's assume that M >= N and the columns are linearly ...
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Technique/solver to handle a constraint of degree 5

I am working on an optimization that involves an equality constraint with an unknown raised to the power 5. \begin{equation} \pi{_{i}}- \pi{_{j}} = a.\dot{V}{_{ij}}^2.\frac{l_{ij}}{d{_{ij}}^5},...
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Gradient, scalar product and constraints [closed]

I've a function $f=f(x,y,z)$ and a given constant vector $\vec A$. Then, there is an equation $\nabla f \cdot \vec A=\rm something$ that for me has a physical meaning. I'd like to write the same ...
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Counting free variables/degrees of freedom from constraints in a function

Say I have some function $f(a,b,c,d,e,f,g,h) = abcedefgh$ with 8 free variables. Normally, I put some constraint, like $a = c$ or $\frac{a}{e} = \frac{b}{g}$, so each equality gives 1 constraint. ...
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1answer
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Stuck at making a constraint for given LP problem where a machine can make one product or the other.

This is the text for following linear problem: In one factory there is a production machine which is available 170 hours a month. Using this machine it is possible to produce 50 pieces of product A ...
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Optimising class assignment based on test score and class choice?

Suppose some students enrol themselves onto a course. The course has 6 available classes, A,B,C,D,E,F, each at a different time and/or day. All the classes have the same upper limit on size, say some ...
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Constrained optimization: first order necessary condition (Lagrange multipliers)

Why does the first order necessary condition for constrained optimization require linear independence of the gradients of the equality constraints at the local minimum point? Liberzon's "Calculus ...
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Boolean condition as constraint of a continuous optimisation problem?

Let $\Theta$ be the space of real invertible $n\times n$-matrices, and for $\theta\in\Theta$ write $\theta_i\in\mathbb{R}^n$ for the $i^{\mathrm{th}}$ row of $\theta$, i.e. $\theta=(\theta_1|\cdots|\...
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1answer
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Error in solving isoperimetric problem with first integral formulation

In order to solve the isoperimetric problem, I am extremising the functional: $$A[x,y] = \frac{1}{2}\int_0^{2\pi}(x \dot y - y \dot x) \;dt\tag{1}$$ where $x = x(t)$, $y = y(t)$, and the Lagrangian is:...
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Conditional constraints for continuous variables

How could we model conditional constraints for two continuous variables? Suppose the two variables are: $$x\geq0$$ $$y\in\mathbb R.$$ The conditions are: if $y>0$, then $x>0$ and if $y\leq0$, ...
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Task Assignment With Constraints

I'm working on a problem and am curious if there is already a known algorithm that solves it. Here is my problem: I have $n_t$ tasks where each task belongs to one of $n_c$ categories $c_1, c_2, ..., ...
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2answers
69 views

Elementary issue with constrained optimization

I am trying to solve $$\begin{array}{ll} \text{extremize} & f(x,y) := x^2+3y\\ \text{subject to} & \dfrac{x^2}{4} + \dfrac{y^2}{9} -1 = 0\end{array}$$ I cannot understand why I am able to find ...
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When does unconstrained optimum being infeasible mean constraint binds?

Let $a,b,c,d$ be non-negative vectors in $\mathbb{R}^n$. Consider the non-linear program: $$ \max_p \frac{a'p}{b'p}\\ s.t.\;0\leq p\leq 1,\\ \frac{a'p}{b'p}=\frac{c'p}{d'p}. $$ My approach is: I'm ...
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Relaxing constraints of min-max optimization problem

I have the following optimization problem \begin{array}{l} (P_1) ~~~~~~ \min\limits_{x \in \mathbb{R}^d} \max\limits_{y \in \Delta} \sum\limits_{i=1}^N y_i f_i(x) \\ \end{array} where the set $\Delta$ ...
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Condition to make a matrix symmetric and feasibility of the solution

Given a diagonal matrix $\mathbf{D}$ of size $R^{n \times n}$ and a full rank matrix $\mathbf{B}$ of size $R^{n \times n}$, what would be the conditions on $\mathbf{B}$ such that $\mathbf{DB}$ is ...
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Constraints giving infeasible solutions for a quantity allocation problem

I have a problem of quantity allocation model where I have 4 sources and 3 destinations. I have formulated this problem as linear programming model. I have the cost from taking materials from sources ...
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1answer
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Solve $Ax=b$ subject to $x=Cy+d$

$Ax=b$ is a linear system of equations in which $A$ is a square invertible matrix. However, I would like to approximately solve $Ax=b$ in which the constraint $x=Cy+d$ is exactly enforced. $C$ is $m\...
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Optimization Problem with Constraints

I am trying to solve the following optimization problem \begin{align} \min_{X_{1},X_{2},y_{11},y_{12},y_{21},y_{22}} \; \; p_{1}X_{1}+p_{2}X_{2}& \\ \text{s.t}\; \; X_{1}^{\beta} &= y_{...
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Probability with constraints

Four numbers $a, b, c, d$ are independent random variables and given by the uniform distribution in $[-1/2, 1/2]$. $t$ is a fixed constant greater than $0$. I would like to compute the probability ...
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maximize a parametric function with a nonlinear parametric constraint

I have the following problem: I have two variables, "$n$" and "$d$", and two parameters, "$t$" and "$F$". $$n\ge 1, 0\lt d<\frac{1}{2n}; F>0, t>0$$. I ...
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How to define constraint for consecutive elements?

I have a sequence denoted by $\vec{x}=(x_1,\dots,x_M), x_i\in\{1,\dots,N\} $. The sequence $\vec{x}$ can later be encoded into another sequence $\vec{c}=(f(x_1),\dots,f(x_M)), f: \{1,\dots,N\} \mapsto ...
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2answers
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K-Median constraints - FLP Linear Programming

I'm attempting to solve a problem using linear programming. I have been using this resource (k-median problem midway down the page) as a reference. I have successfully implemented a first attempt ...
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Have graph recovery problems been studied?

I’m interested in the minimum number of edges required to specify a graph $G$ uniquely when $G$ is known to satisfy some property. For example, if we know that $G$ is a cycle graph on $n > 3$ ...
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Proof that the difference between the input and output of the proximal operator belongs to the subdifferential

The proximal operator $\text{prox}$ is defined as follows, for a function $f$, at point $x$: $$\text{prox}_f(x) = \underset{u}{\text{argmin}} (f(u) + \frac{1}{2} ||x-u||^2)$$ The subdifferential $\...
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Penalty Function for XOR Gate.

I have been reading a paper on Gates for Adiabatic Quantum Computing. The paper consists of penalty functions for different classical gates like AND, OR, XOR, etc. Can someone explain how to get the ...
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constrained polynomial equation system arising from a possibility to probability transformation

I have defined a probability-to possibility transformation as follows: $\pi_1 = \sum_{j=1}^Np_j$ $\pi_2 = p_{2}^2\cdot \frac{1}{p_1}+\sum_{j=2}^Np_j$ $\pi_3 = p_{3}^2\cdot (\frac{1}{p_1}+\frac{1}{p_2})...
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$a \geq b \geq c \geq 0$ and $a + b + c \leq 1$. Prove that $a^2 + 3b^2 + 5c^2 \leq 1$.

Positive numbers $a,b,c$ satisfy $a \geq b \geq c$ and $a + b + c \leq 1$. Let $f(a, b, c) = a^2 + 3b^2 + 5c^2$. Prove that $f(a, b, c) \leq 1$. One observation is that the bound is met: $f(1, 0, 0) = ...
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1answer
63 views

Converting a basic MILP into LP

I have a MILP looks like following: $$ min \sum_i capacity[i] * weight[i] $$ where i=[Mid_North_1, Mid_North_2, North_Mid_1, North_Mid_2] Basically I got 2 sites ...
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1answer
48 views

A discrete optimal control problem

I've been looking into control theory recently, but have been struggling to find ways to solve a particular question of mine. It seems to be formulated as a discrete optimal control problem with ...
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1answer
26 views

Equivalence of multiplying matrix inequality constraint by inverse

I'm trying to understand the conditions for which $$ A x \leq b \Leftrightarrow x \leq A^{-1}b$$ is true. Let's assume that $A$ is a non-singular, $x$,$b\in \mathbb{R}^n$. I have come across an ...
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2answers
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Integral Optimization Problem with inequality constraint

I would like to find the function $p(x)$ that maximizes the integral $$I(p) = \int_{0}^{L}p(x)q(x)dx$$ subject to constraints $$\int_{0}^{L}p(x)dx=1$$ $$\int_{0}^{L}xp(x)dx=1$$ $$p(x) \geq 0 \;\; \...

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