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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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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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 $x^2 > x$ for all $x \in \mathbb{R}$, $f$ is bowl-ish with a minimal value in the bottom. ...
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20 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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1answer
85 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
21 views

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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1answer
25 views

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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2answers
115 views

How does one generate recognizable point-patterns on a plane?

I've recently learned that some smartpens (e.g. Livescribe) have a camera in their front part. They film the paper. You have to use special paper which looks as if somebody made a lot of tiny holes ...
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1answer
36 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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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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1answer
13 views

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
319 views

Gradient-descent and Hidden Markov Models

I would like to use gradient-descent to fit the parameters of a simple 2-state HMM. This paper Levinson, S. E., Rabiner, L. R. and Sondhi, M. M. (1983), An Introduction to the Application of the ...
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2answers
2k views

Optimization of the surface area of a open rectangular box to find the cost of materials

A rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of the box is twice its width. Material for the base costs ten dollars per square meter and for the ...
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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
29 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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1answer
8 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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0answers
67 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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1answer
29 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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1answer
29 views

A circular field encloses maximal area for minimal perimeter [duplicate]

Suppose a farmer has a certain length of fence, $P$ and wished to enclose the largest possible area. What shape area should the farmer choose? Answer is "circle".But, how is it derived? MY TRY: My ...
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0answers
23 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
14 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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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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2k views

Linear Programming: Three variable graphical solution

A small bank offers three type of loans: housing loans at $8.50$% interest, education loans at $13.75$% interest rates, and loans to senior citizens at $12.25$% interest. Further, it needs to ...
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1answer
20 views

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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Least squares optimization problem, KKT conditions and derive expression for $x^*$ [on hold]

minimize $\|Ax-b\|^2$ subject to $Kx=d$ where $K$ is a constant a) Give the KKT conditions b) Derive an algebraic expression for the optimal solution $x^*$ c) Determine the point $x^*$ that ...
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1answer
61 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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1answer
13 views

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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Can a dynamic programming problem be transformed into a linear algebra problem?

Here is a simple standard economic problem: Let Robin Crusoe have an endowment $w_0$ of lembas bread. She is immortal and discounts at a rate of $\beta$ per period. Each period (from $t=0$ on) she ...
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My attempt regarding finding critical ponts of $(\cos x)(\cos y)(\cos(x+y))$

Given this problem Restrictions on $x$ any are that $x\in[0,\pi]$ , $y\in[0,\pi]$ I have $f_x=-(\cos y)({\sin(2x+y))}--------*$ $f_y=-(\cos x)(\sin x+2y)-----------**$ So from $*$ I get either ...
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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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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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1answer
19 views

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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Setting up the Bellman equations for dynamic programming

I have the following question I want to understand. The owner of a chain of three grocery stores has purchased five crates of fresh strawberries. The estimated probability distribution of ...
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28 views

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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57 views

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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80 views

Optimal allocation in network

We want to analyse specialization matters in a given network (N,g). Nodes represent individuals that can produce goods and services (just like in our usual economy) and that can be consumers too. ...
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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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Optimize to Find the Mahalanobis Distance to Minimize the Term

I have an optimization problem defined as following: Assuming we have a data set $ { \left\{ \left( {x}_{i}, {y}_{i} \right) \right\}}_{i = 1}^{N} $ where $ {x}_{i} \in {\mathbb{R}}^{d} $ and $ ...
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2answers
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Solving a matrix equation using numerical optimization

To my knowledge, if $A \in \mathbf{S}^n_{++}$, then given any $b \in \mathbb{R}^n$, the system of linear equations $Ax = b$ has a unique solution $x^* \in \mathbb{R}^n$. Moreover, the solution $x^* ...
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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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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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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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Is the ratio trace problem convex?

I have a ratio trace problem described as follows: $\arg\max_{w} \text{trace}((w^tAw)\cdot \text{inv}(w^tBw))$, where A and B are full rank matrices. This problem can be solved via generalized ...
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1answer
27 views

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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What's wrong in this dual derivation?

I have a function in the form \begin{align} f(q,M)=\sup_{0\leq \alpha \leq 1} -\alpha^T (R\odot M)\alpha+\alpha^Tq \end{align} which is a dual of a minimization problem, where $R$ and $M$ are ...
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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 ...
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88 views

Convex Optimization Closed Form Solution

I'm currently studying for my exame in convex optimization. It covers problem formulation, first and second order conditions of optimality, unconstrained optimization, constrained optimization with ...
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55 views

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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276 views

Intuition behind accelerated first-order methods

$\newcommand{\prox}{\operatorname{prox}}$ $\newcommand{\argmin}{\operatorname{argmin}}$ Suppose that we want to solve the following convex optimization problem: $\min_{x \in \mathbb{R}^n} g(x) + ...
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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 ...