# Questions tagged [constraints]

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

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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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...
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 \;\; \...