Optimization is the process of choosing the "best" value among possible values. They are often formulated as questions on the minimization/maximization of functions, with or without constraints.

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How to find out if point is local Maximizer or local Minimizer ? Lagrangian is given

The Lagrangian is: $L(x,\lambda) = x_1x_2-2x_1-\lambda (x_1^2-x_2^2)$ Taking the derivatives and setting it equal to zero gives: $x_2-2\lambda x_1-2=0$ $x_1+2\lambda x_2=0$ $x_1^2-x_2^2=0$ The ...
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When exactly are quadratic objective functions polynomial time solvable

I'm considering quadratic programming problems of the form: $$ \max x^tQx+Bx$$ subject to the linear constraint $$ Ax \le b $$ I read that if is the case that $$ x^tQx + Bx \ge 0 \ \forall x$$ or ...
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Finding conditional extrema with trig functions

Find the conditional extrema of $$f(x,y)=\cos^2x+\cos^2y,\quad g(x,y)=x-y+\frac{\pi}{4}=0.$$ I have a problem with finding a solution to this problem. Using Lagrange multipliers i come up with a ...
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Optimization problem: maximise horizontal distance

Fifteen men are placed on a Dead Man's Chest in a rectangular pattern, with each man distant $a$ from his neighbours,thus: The average weight of the men is $w$, and the heaviest man weighs no ...
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Find min & max of $f(x,y) = x + y + x^2 + y^2$ when $x^2 + y^2 = 1$

Problem: Find the maximum and minimal value of $f(x,y) = x + y + x^2 + y^2$ when $x^2 + y^2 = 1$. Since $x^2 > x$ (edit $x^2 \geq x$) for all $x \in \mathbb{R}$, $f$ is bowl-ish with a minimal ...
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32 views

Maximum likelihood of fraction

To maximize the log likelihood of my parameters, I need to find the argument that maximizes the following function: $$\sum_{\substack{i,\,j\\ i \neq j}} n_i \log(x_i) + n_j \log(x_j) - (n_i + n_j) ...
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1answer
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Is the given function convex? Does the point satisfy the FONC?

minimize $2x_1^2+x_2^2-2x_1x_2-4x_2$ subject to $x_1x_2\leq 4$ Is the objective function convex? Does the point $(1,4)$ satisfy the FONC for a local minimizer? Do optimal solutions of this given ...
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Learning to solve complex inequalities in many variables

below is a very specific inequality problem. I would like to know how to solve it so I can apply it to more complex problems. The equations are as follows: $$3.5x−2.5y−3z=A$$ $$−7.5x+3.75y+5.25z=B$$ ...
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26 views

Does given point satisfy FONC?

minimize $4x_1^2+2x_2^2-4x_1x_2-8x_2$ subject to $x_1+x_2\leq 4$ Does the point $(2,2)$ satisfy the FONC for a local minimizer? The gradient of the objective function is $\nabla f = ...
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One definition of strong convexity (from textbook of Prof. Bertsekas in 2015)

In strong convexity, there are a few definitions, one of them is: $f$ is strongly convex over $\mathcal{C}$ with coefficient $\sigma$ if $\forall x,y \in \mathcal{C}$ and all $\alpha \in [0,1]$, we ...
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1answer
98 views

quantum mechanics violate Bell's inequality

I have this function $$ \begin{aligned} F\big(\theta_a,\theta_b,\phi_a,\phi_b\big) = \ & – \big[\cos \theta_a \cos \theta_b \big] – \big[\sin\theta_a \sin\theta_b \sin\phi_a \sin\phi_b\big] \\ ...
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1answer
11 views

rectangular paddock, dimensions, maximise area it encloses

Having trouble trying to work out a question which involves finding a function to graph evidence of the correct answer, any advice would be greatly appreciated. I am struggling with part 'b' a lot, ...
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31 views

Minimize distance between polynomials, of a certain form, with Laguerre polynomials

A typical problem that I may encounter on an upcoming test looks like this: Find the polynomial $P(x)$ of a degree less than or equal to three that minimizes $$\int_0^\infty (x^4 - ...
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28 views

Solving large system of Linear equations

I am trying to solve an optimization prob of the below form: $$ \min \sum_{k=0}^{n} p_k$$ subject to : $$0 \leq p_k \leq p_{\max}$$ $$ g_k p_k \leq I_t$$ $$g_k p_k - \eta_k \sum_{j \neq k} p_j ...
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1answer
32 views

L1 norm differentiablility

I am trying to understand the Least Absolute Deviation algorithm, which basically is min l1-norm(z) subject to z=Ax-b I want to understand how is the l1-norm ...
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29 views

Optimization problem involving semidefinite matrix variable that is constrained to be a tensor product

I would like to solve the following optimization problem. With scalar $R$ and nine (mutually orthogonal) $9$-dimensional column vectors $\vec v_i$ all given ($\vec v_i\!'$ is the row vector Hermitian ...
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1answer
16 views

How to interpret multiple critical points (from Lagrange multipliers) that all give a maximum value

If I have 6 critical points, 3 of which give the same maximum possible value of a function f(x,y,z), subject to a constraint g=c, is there something more to say about this solution -- or we just ...
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1answer
28 views

Minimizing long equation with hyperbolic functions

In physics book that I am reading it is said that minimizing the expression $$\phi = - N T k \log (2 \cosh(H \beta)) - \frac{J N}{2} z \tanh^2(H \beta) + H N \tanh(H \beta) $$ with respect to $H$ ...
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79 views

The maximum value of PA.PB.PC

Let A,B,C be the vertices of a triangle inscribed in a unit circle, and let P be a point in the interior or on the sides of the triangle ABC. Then the maximum value of (PA)(PB)(PC) equals to? I could ...
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50 views

Solve this set of Lagrange multiplier equations,

I'm trying to solve $$(yz,xz, xy) = (\lambda\frac{2x}{a^2},\lambda\frac{2y}{b^2},\lambda\frac{2z}{c^2})$$ with the constraint equation $$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1$$ ...
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What is the maximum value of work done by this force field?

An object moves in the force field $F=yz\hat{i}+zx\hat{j}+xy\hat{k}$ starting at the origin and ending at some point $A(\xi,\eta,\zeta)$ that lies on the surface ...
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Optimizing space for many shapes within an irregular shape

So let's say in a state, there are 50 schools dispersed throughout, given by Latitude Longitude points. How would we create distinct zones that optimize the space around each school? The goal is to ...
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How to find the point in convex set $C$ that is closest to $y\notin C$?

How to find the point in convex set $C$ that is closest to $y\notin C$? $C=\{ x\in \mathbb{R^2}:(x_1-1)^2+(x_2-1)^2\le1 \}$ and let $y\notin C $ but $y\notin \mathbb{R^2} $.
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Intuitive way to find the minimum surface in $\mathbb R^3$?

Suppose we have two arbitrary closed curves which intersect neither each other nor themselves. By intuition, I guess that the minimum surface ending at boundaries is unique and it is achieved by this ...
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What does it actually mean if a cost function is differentiable?

I am just learning about optimization, and having trouble understanding the idea behind differentiating cost functions. I have read that for standard optimization problems, the cost function needs to ...
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Determine Critical points in optimisation problem

So I have this problem where I am supposed to calculate the max and min value of a function $f(x,y)=x+2y$ restricted by the disk $x^2+y^2\le 1 $. I have calculated the $df/dx $ and $df/dy$ and they ...
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Single nonzero value constraint formulation in linear programming problem statement

I'm trying to write a linear programming problem statement. Values of the solution vector have a bound constraint: $0 \leq x_i \leq 1$. Another constraint is that if we take a predefined subset of ...
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Optimization problem shortest path distance and critical node detection problem (interdiction).

I am trying to formulate this optimization problem, max $d_{ij}$ where $d_{ij}$ is the shortest distance between active nodes i and j. However my problem is connecting my decision variable with the ...
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Can this be expressed in terms of linear constraints?

I'm attempting to find a matrix $X$ that minimizes some function $f(X)$ subject to the constraint that $$ X=W A Z $$ where $A$ is a given non-negative matrix with rows that sum to 1, and $W$ and $Z$ ...
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Use graphical methods to solve the linear programming problem. Maximize:

Use graphical methods to solve the linear programming problem. Maximize: $z= 4x+2y$ subject to : $x-y\le 7$ $19x+12y\le 228$ $18x+18y \le 324$ $x\ge 0,y\ge 0$ the max is ?? when x= ?? and ...
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52 views

Find the minimum of a function for only positive values of the vector variable

Let variable vector $\vec{q}$ of size $m\times1$, and its diagonal counterpart $m\times m$ matrix $Q=diag(\vec{q})$, for some $m\in\mathbb{N}$. Define fixed parameter $n\times1$ vectors $\vec{p}, ...
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Relaxation of non-convex QCQP with one quadratic and one linear constraint

According to Boyd we know that a non-convex QCQP problem with one quadratic constraint has strong duality with the relaxed SDP or Lagrange counterpart. (check "Convex Optimization" by Boyd, Appendix ...
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Calculating min/max of a multivariate function on a region

This video shows an example of how to find the absolute maxima and minima of the function $f=xy+y^2$ at the region $\{(x,y):|x|\leq1,|y|\leq2\}$. I understand why he set $f_x, f_y$ to $0$, checked ...
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super-additive, sub-additive, and shapely value limitations?

I am working on the coalition formation. Most of the scientist used concept of shapely value for distributing the utility among the members of coalition. Up to my understanding, shapely value is good ...
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116 views

Maximization of sum of functions

Let $w,a\in R^n$, and $B\in R^{n\times n}_{++}$ (the set of $n\times n$ positive definite matrices). We know that the following function (which is a specific form of the Rayleigh quotient) has a ...
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A question about a proof in nonlinear programming book

I have a question about the proof of Proposition 1.2.1 (Stationarity of limit points for gradient methods) in the nonlinear programming book (2nd edition) by Bertsekas. At the beginning of the proof ...
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Max value of trignometric function $\sin \left(x+\frac\pi6\right)+\cos \left(x+\frac\pi6\right)$

Question: The maximum value of $\sin \left(x+\dfrac{\pi}{6}\right)+\cos \left(x+\dfrac{\pi}{6}\right)$ is at what value of $x$. I solved the problem by setting the slope of the function to zero and ...
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38 views

Curve fit minimizing the sum of the deviation

I'm fitting a curve taking the smaller sum of deviations for each parameter tested, the smaller sum returns me the parameter that gives the best fit. Here is the algorithm for a test $f(x, ...
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The dual function of composite functions

Given $X$ $Y$ are two finite dimensional Hilbert space. Let $K$: $X\to Y$ be linear and $F$: $Y\to \mathbb R^+$ is convex. Let us use $F^\ast$ to denote the dual (conjugate) function of $F$. Recall $$ ...
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Calculating amount of cubes that fit in a sphere

I know that the problem of finding out how many spheres can fit in a cube is a commonly asked and well documented one, but I am struggling to find anything on the inverse of the problem, namely: ...
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Sensitivity Analysis: Calculating Allowable increase/Decrease for a Binding Constraint

Let say we have the following equations: Objective Function = $7T+5C$ Contraints $3T + 4C \le 2400$ $2T + C \le 1000$ $C \le 450$ $T\le 100$ How would we calculate the allowable increase and ...
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Taking derivate wrt a vector

I'm trying to read through Wiki's description of the Levenberg-Marquardt algorithm. I've taken linear algebra, but I've always been fuzzy about taking derivatives with respect to a vector and just ...
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Distance between a plane and set of points

Suppose $m$ data points belonging to a class represented by matrix $A$. Therefore, the size of matrix $A$ is $m\times n$. In addition, suppose $w\cdot x + b=0$ be equation of a plane in ...
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Methods to Minimize Functions and Integrals over $\mathbb{N}$.

In a paper I'm writing, I have to minimize a messy function $f(\mu,n)$ where $\mu \in \mathbb{R}$ and $n \in \mathbb{N}$. That is, given $\mu \in \mathbb{R}$, I need to minimize the one variable ...
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65 views

Is this a game theory problem or optimization problem?

Consider a problem that looks for a $x$ that can make the following problem into some equilibrium state (similar to an equilibrium solution to a min-max problem in game theory) $$ \max_x f(x)$$ $$ ...
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Shaking a box of rocks (Optimal Packing)

My coworker was telling me that when he plants seeds on his farm, he puts them all in a large container on the tractor and after a period of just driving, the seeds are more densely packed than when ...
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Maximum of uniform random variable [closed]

Suppose I want to compute: $\max \{E[2 \min(\theta, I)]-I\}$ where $I$ is the choice variable and $\theta$ is a uniform variable in [0,1]. How should I do it?
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Fit polynomial function using experimental data (least squares)

I want to fit the polynomial function $f(x) = \alpha_0 +\alpha_1 x +\alpha_2 x^2 $ using given data such that the errors $y_c-f(x_c)$ are minimized (least squares). Obtained is the experimental ...
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Finding extrema of function of three variables

So i have to study this function and find out if there are any local or absolute extrema : $ f:\mathbb{R}^3 \rightarrow \mathbb{R} :$ $$ f(x,y,z)=2-\left(z-\sqrt{x^2+y^2}\right)^2 + ...
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If this problem is not unbounded, what's wrong in this dual derivation?

In a paper with 100 citation, Robust Support Vector Machine Training via Convex outlier Ablation, a convex relaxation is used. In this paper, a form of robust svm proposed: \begin{align} \min_{0\leq ...