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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Confounding Lagrange multiplier problem

Optimize $f(x,y,z) = 4x^2 + 3y^2 + 5z^2$ over $g(x,y,z) = xy + 2yz + 3xz = 6$ According to the theorem the gradients must be parallell, $\nabla f = \lambda \nabla g$, so their cross product must ...
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Numerical nonconvex optimization problem

I have numerical data for the mapping $w:\mathcal{S}^{2+}\to\mathbb{R}$, where $\mathcal{S}^{2+}$ is $\{\mathbf{x}\in\mathcal{S}^2:x_3\ge0\}$, the 2-hemisphere on or above the $x_1-x_2$ plane. I ...
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Alternating Direction Method of Multipliers (ADMM) application

$\newcommand{\argmin}{\operatorname{argmin}}$ Recall, that ADMM algorithm solves the problem of the form: $\min \text{ } f(X) + g(Z)$ $\text{s.t. } AX + BZ = C$ where $X$, $Z$ and $C$ are real ...
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Gradient descent vs. Newton's method — which one requires more computation?

Broadly speaking, when numerically minimizing a d-dimensional objective function: Gradient descent generally requires more iterations, but each iteration is fast (we only need to compute 1st ...
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31 views

Evolutionary algorithm

Can someone provide me a good reference for the CMA-ES algorithm? I'm new in the world of optimization and just reading the author reference doesn't help me a lot. I know the basic idea of a genetic ...
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15 views

“Rank-K Correction” of a matrix and significance?

Today my studies led me to read about the matrix inversion lemma, which Wikipedia introduces as follows: In mathematics (specifically linear algebra), the Woodbury matrix identity, named after Max ...
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12 views

How to run a number of tests with differing transitions?

I believe this would be the correct exchange to ask this question. I have a black box with 3 dials, a button, and a display. Each of the dials can be spun to be set to a number on that dial. The ...
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157 views

Maximizing “log det + log sum exp” function

I'm trying to find a numerical solution to the following optimization problem $$ \text{maximize } f(M) = \frac{1}{2} \log \det(M) + \log \sum_{i=1}^n \exp \left\{ - \frac{1}{2} x_i^T M x_i + a_i ...
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45 views

Finding the minimum distance from the origin to the surface $xyz^2=2$

This was an old exam question I was looking at for a friend, although it's been a while since I've done this stuff: Q. Find the shortest distance from the origin to the surface $xyz^2=2$. I ...
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46 views

How to minimize the square of $\sum b_i x_i$ where each $b_i$ can be either $0$ or $1$?

Would you please help me solve the following problem where $b_{ij}$ is my decision variable that must be determined and all other parameters are known. $$\min \left(\sum_{i=1}^n b_i x_i\right)^2$$ ...
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104 views

Knuth's Sandwich Theorem: requesting proof clarification

The question is about F6 of Section 8 ("Elementary facts about cones") in Donald Knuth's Sandwich Theorem (http://arxiv.org/pdf/math/9312214.pdf). He claims to prove $(A \cap B)^* = A^* + B^*$ when ...
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Optimization problems on the circle

Consider the optimization problem $$ \min_{x \in \mathbb{R}^2} x^{\top} P x + q^{\top} x$$ subject to: $$ A x = b, \ x \in X, \ x_1^2 + x_2^2 = 1$$ where $X$ is compact and convex. Then ...
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37 views

Maximum Area of a Triangle when 1 Side, Perimeter Known

This is an example of a "quantitative comparison" question the GRE would test. Suppose the following information is known: one side of a triangle has length 12 the perimeter of the triangle is 40 ...
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20 views

Smallest linear combination of a set of vectors

I'm searching for an algorithm to accomplish a (hopefully) simple task. If I have a set of vetors, (e.g. $\left( ...
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91 views

Conditional extreme value of a function

Let $x,y,z$ be the positive real numbers, if $x^2+y^2+z^2=1$, then how can we find the minimal value of this function $f(x,y,z)=\dfrac{xz}{y}+\dfrac{yz}{x}+\dfrac{xy}{z}$.
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50 views

If a continuous function on $\mathbb R$ satisfies $f(x)\ge x^2$, it attains its minimum

This question is from my homework and I don't know how to prove it. Let $f(x)$ be a continuous function at $\mathbb{R}$. prove that if $f(x)\geq x^2$ to every $x$ in $\mathbb{R}$, then $f(x)$ ...
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78 views

Proof of a matrix is positive semi-definite

For $\ i = 0, 1, \cdots m$, $f_{i}(x): R^n \rightarrow R$ is defined to be $$ f_i(x) = x^TQ_ix + 2p_i^Tx + r_i $$ , where $Q_0 \cdots Q_m$ are real symmetric matrices, $p_0 \cdots p_m \in R^n$, and ...
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33 views

How to linearize a quadratic objective function with linear constraints?

I have an optimization problem that I'm working on. The objective is defined as follows: $Maximize: c_i\cdot w_i \cdot x_i - d_i \cdot y_i \cdot \delta_i $ subject to some linear constraints where ...
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45 views

Maximizing business profit [closed]

150 computers sold at 50 dollars (C) each. A loss of 7 customers (N) is expected for every price increase of 10 dollars. Other costs: 3000 dollars for wages 0.1 x N^2 for fuel (C is cost, N is ...
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Dual problem of SDP

Suppose we have the following optimization problem: \begin{array}{l} \mathop {\min }\limits_{{\bf{X}},{\bf{x}}} \,\mathrm{Tr}\left( {{\bf{XA}}} \right) + 2{{\bf{a}}^H}{\bf{x}} + b\\ ...
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Ellipsoid-Sphere Intersection [duplicate]

Suppose I have an Ellipsoid given as: $$\frac{(x-x_2)^2}{a^2} + \frac{(y-y_2)^2}{b^2} + \frac{(z-z_2)^2}{c^2} = 1$$ And a Sphere $$(x-x_1)^2 + (y-y_1)^2 + (z-z_1)^2 = R_1^2$$ If they intersect ...
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27 views

How many max/min/saddle points of $f(x, y)$ on a region?

Consider the function $f(x,y)=2xe^x\sin y$ on the region $\{(x,y) \mid -\frac{\pi}{4} \leq y \leq \frac{3\pi}{4}\}$. How many maximums/minimums/saddle points are there? I honestly have no idea how ...
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Global Maximums and Minimums

My book states: "It is also true that if $x^*$ is an interior point and: a global maximum of $f$ , then $d^2f(x^*)$ is negative semi-definite. a global minimum of $f$ , then $d^2f(x^*)$ is positive ...
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Armijo conditions vs Reduction Conditions in Non-Linear Line Search

Overview Line search typically consists of four stages: Direction: Search direction Initial Step Size: length to search along the line on the first sub-iteration Bracket: find an interval along the ...
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Quadratic programming using Python

guys I'm trying to solve quadratic programming problem with constraints. I know how to solve simple quadratic problems using scipy.optimize like following: Define objective function as F = ...
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algorithm to find the root of a real-valued function $f$

I see in a book the following algorithm to find the root of a real-valued function $f$ $$ \theta_{n+1} = \theta_{n} + \epsilon f(\theta_n); \epsilon >0 $$ with the condition that the initial ...
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$y_{t+1}$ where $t=t_0,t_1,\dots$

Objective function: min $\frac{1}{2}\sum_{t=t_0}^{\infty}y_t ^2$ $y_t=E_t y_{t+1}-0.5(i_t-2)+u_t$ where $t=t_0,t_1,\dots$ $u_t=0.5 t_{t-1}+\epsilon_t$ where $\epsilon$ is i.i.d and zero expectation. ...
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48 views

minimizing sum of different least squares?

Can we write the minimization problem: $$\operatorname{min}\limits_{x\in\mathbb{R}^n}\sum_{i=1}^{n}\|C_i x-b_i\|_2^2$$ as a least square problem?
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Equilibrium in a first-price common value auction with a normal distribution?

Does anyone know of a paper that derives equilibrium bidding strategies in a first-price common value auction when signals are assumed to come from a normal distribution?
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Maximising a function under a constraint

Let $$f(x,y,z) = 4x+2y+5z^2 \text{ and } A=\{(x,y,z) \in \mathbb{R^3} ; \, x^2+y^2+z^4 \leq 5 \}.$$ Find the maximum of $f$ on $A$. My question is the following: How do I prove that the maximum must ...
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25 views

How to deal with non-existent derivatives in Lagrangian?

I am stucked at a detail in a constrained optimization problem: Question Assume that the objective function is continuous on its domain $D$, but at some points $Z \subseteq D$ it is not ...
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62 views

Finding Shortest distance between a Sphere and Ellipsoid?

Suppose that ,I have a Sphere and an ellipsoid as Sphere: $(x-x_1)^2 + (y-y_1)^2 + (z-z_1)^2 = R_1^2$ Ellipsoid: $\large\frac{(x-x_2)^2}{a^2} + \frac{(y-y_2)^2}{b^2} + \frac{(z-z_2)^2}{c^2} = 1$ ...
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34 views

Optimization Software

I am currently trying to optimize an equation that contains 4 variables. It is nonlinear and non-convex. In mathematica, to optimize this equation, I take the partial derivative of each variables, ...
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25 views

Coordinate descent with constraints

Coordinate descent is a powerful method for solving optimization problems like $$\min_x \tfrac{1}{2}x^T A x + b^T x + \lambda ||x||_1$$ where $A$ is symmetric and positive definite, $\lambda>0$ ...
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Objective function: $\min \frac{1}{2}\sum_{t=t_0}^{\infty} y_t ^2$

Objective function: $\min \frac{1}{2}\sum_{t=t_0}^{\infty} y_t ^2$ $y_t=E_t y_{t+1}-0.5(i_t-2)+u_t$ where $t=t_0,t_1,\dots$ $u_t=0.5 t_{t-1}+\epsilon_t$ where $\epsilon$ is i.i.d(independent and ...
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Baseline predictors model

I implemented baseline predictors model (like it is told in Recommender systems handbook pp 148-149). b_ui = mu + b_i + b_u where mu is overall average rating ...
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Alternative to Hungarian Algorithm to determine minimum cost?

Is there a graphic calculator (CAS technology) method to solve minimum cost problems/allocations that are normally completed with the Hungarian Algorithm... Hungarian Algorithm is time consuming, ...
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27 views

resources about sparse global constrainted optimization

Please recommend a good resources (books/articles/software) about sparse global constrained optimization?
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Biggest subset of $\{1, 2 … 1000\}$ such that difference between any pair of elements $\neq 4, 7$

The problem, as stated in the title, is to find the maximal size of a subset $V$ of $S = \{1, 2, ... 1000 \}$ such that no two elements of $V$ have a difference of 4 or 7 between them, i.e. $x \in V ...
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How to load warehouse pallets efficiently?

Assume that we would wan't to develop a warehouse management system, which picks up plastick boxes and stacks them on a pallet. A pallet has a maximum of 5 vertical box stacks and the maximum height ...
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Critical points, minima and maxima of a constrained optimisation problem

$f(x_1,x_2,x_3) = x_1 x_2 + x_2 x_3 + x_3 x_1$ with the constraint $x_1+ x_2 + x_3 = 1$. Now I test the critical points by taking $\nabla f = 0$, hence at $(x_2 + x_3, x_1 + x_3, x_2 + x_1) =0$. This ...
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Find the point where the function $C(t) = 0.24t/(t^2 + 5t +4)$ attends its maximum [closed]

The concentration, $C$, in micrograms per deciliter, of a drug in a patient's bloodstream is given by the equation, $C(t) = \dfrac{0.24t}{t^2 + 5t +4}$ , where $t$ is the number of hours after the ...
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Are there some functions that cannot be optimized using calculus?

I've been working on a project to maximize a functions output using a genetic algorithm. However, from the limited calculus I know I thought there were methods to find the maximum of a mathematical ...
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squaring the equality constraints

When creating an unconstrained optimization problem from an equality constrained one, the usual way to build the Lagrangian, is by adding a term consisting of a multiplier, multiplied by the equality ...
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Deriving estimators for the parameters a and b that minimize the random error - setting up linear regression variables?

I'm reviewing old notes, and I know I solved this way back when, but can't remember how to know: Consider the simple linear regression model: $$Y_i = a + bX_i + \epsilon_i$$ where $Y_i$ is the ...
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Steepest descent direction with surface constraint (geometry problem)

Let say I have a function $f(x,y,z)$, defined on a surface by the level curve $g(x,y,z)=c$. I want to know what is the direction of the steepest descent at a given point, taking into account the ...
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49 views

Minimizing the cost of production by choosing between two options with different resource demands

Suppose that the firm has two possible activities to produce output. Activity $A$ uses $a_1$ units of good $1$ and $a_2$ units of good $2$ to produce $1$ unit of output. Activity $B$ uses $b_1$ units ...
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34 views

Total number of subsets

I am trying to solve this hackerrank problem https://www.hackerrank.com/contests/101jul14/challenges/colorful-polygon. Not able to understand the editorial. Can some explain how to solve it using ...
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Optimization problems: Finding the optimal path

I'm still trying to get the hang of optimization problems in calculus and I'm looking for a little help. I'm having trouble finding equations to model the following problem: I'm fairly sure I need to ...
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67 views

Why does this vector derivation hold?

I have the following variables/matrices: $$A \in \mathbb{R}^{m \times n} , \quad p \in \mathbb{R}^{n}, \quad \Sigma \in \mathbb{R}^{m \times m}, \quad w \in \mathbb{R}^{m}$$ where $\Sigma$ is a ...