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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Optimum number of disjoint subsets

Be $A \subset D \wedge m \in D \wedge \forall x \in A:x<m$, with $D$ finite, I need to take disjoint subsets $B_n \subset A$ so that $$\left(\sum_{i\in B_n}i\right) \leq m$$ And I would like to ...
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16 views

L1 regularized SVM in Matlab

Minimizing the following SVM formulation \begin{align} \arg\min_{\mathbf{w}}\frac{1}{2}\|\mathbf{w}\|^2_2 \\ \text{subject to } \quad y_i(\mathbf{w}\cdot\mathbf{x_i}) \ge 1 \end{align} can be done ...
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Horn–Schunck method. Explanation of iterative solution

I am reading this paper (explanation of Horn-Shunck method for finding optical flow) and trying to understand it. My stumbling block is obtainig solution of system of linear equations I(x, y, t) ...
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14 views

Reference for gradient descent with unit norm constraint

I faced a non-convex optimization problem with unit norm constraint. I can solve the problem using the gradient descent method and the projection of the gradient onto the tangent plane as in @joriki ...
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1answer
16 views

Max/Min Notation Question

In a paper I'm currently reading it gives alpha to be the following value. $\alpha = \max_t \min_{t_j \in T_N} ||t-t_j||_2$ I am wondering what exactly this means? I have the following code: ...
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multi-objective optimization

I am currently encounterring a optimization problem. The goal is optimize an objective function A and B at the same time. But the problem is that optmizing A will almost always tradoff with B, such ...
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1answer
29 views

Minimal volume of a tetrahedral

I'm unsure how to solve the following problem: Let $\textbf{p}=(a,b,c) \in \mathbb{R}^{3}$ with $a,b,c > 0$. For $\alpha , \beta > 0$ the equation $$\alpha (x-a)+ \beta(y-b) + (z-c) =0$$ ...
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13 views

Determine the maximum cross‐sectional area.

The client wants to maximise the volume of a materials store to be constructed next to a 3 metre high stone wall (shown as OA in the cross section in the diagram). The roof (AB) and front (BC) are ...
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15 views

Goldstein test in nonlinear programming

I'm reading about nonlinear programming and the Goldstein test. Here is the definition from my book: A line search accuracy test that is frequently used is the Goldstein test. A value of ...
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1answer
27 views

determine the maximum cross‐sectional area.

The client wants to maximise the volume of a materials store to be constructed next to a 3  metre high stone wall (shown as OA in the cross section in the diagram). The roof (AB) and  front (BC) are ...
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1answer
19 views

Above what order of magnitude a pure cutting-plane algorithm must be forgotten in favour of branch-and-cut?

Crawling the web on the subject of the cutting-plane algorithm, I have seen everywhere that a pure cutting-plane method cannot be used for numerical instability reasons after some iterations. But do ...
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23 views

How to transform a maximizing objective function which contains a max operator to a standard LP form

My Optimization objective function looks like this: $\max\quad(c_1 x_1 + c_2 \max\{x_2, x_3, x_4\})$ all variables, $x_i$ are binary variables. There are also some linear constraints such as $a_ix_1 ...
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1answer
24 views

What is $s$ in s-energy (eg. Riesz s-energy)

I'm trying to understand fekete problems. There is a variable $s$ and a related concept of 's-energy' [1] [2] [3] [4] that comes up repeatedly when borrowing the concept of potential energy to find ...
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1answer
106 views

find the minimum value of $x^2-6x+9+ \dfrac{64}{x^2}$

Looking for an elegant solution. I can do by brute force, that is finding derivative and double derivative. All Ideas will be appreciated and tried by me.
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1answer
24 views

Simple minimization problem

Suppose we want to execute a program on a processor which can run in three different modes. Each mode can be describe by a pair $(E,\tau)$ where $E$ denotes the energy consumption per cycle (in nJ) ...
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1answer
35 views

Origin of Slater's condition

I've been looking all over the internet to answer this question: Slater's condition is a commonly used to certify that strong duality holds in a convex optimization problem. Although used in many ...
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2answers
25 views

Minima point is a solution point

Consider $$f:\left[0, \dfrac{\pi}2\right] \to \mathbb R$$ defined as $$f(x)=\sup\{x^2,\cos x\}.$$ It is easy to show that $f$ has an absolute minimum point at $x_o \in I$ , but how to show that $\cos ...
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29 views

linear programming, product mix

hi can anyone please help me with this question A company located in London produces two types of telephones: one is a rotary dial telephone and one is a push button telephone. The production manager ...
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41 views

Research: what can I do next if the solution is too long

Sorry for this vague question. Here is my circumstance: I have tried to formulate a problem with an equation, and it is a optimization problem. So I just go ahead, and use mathematica to take the ...
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1answer
27 views

Linear Algebra Question concerning the trace of a symmetric positive definite matrix.

The objective is to minimize the diagonal elements of a symmetric positive definite matrix. The expression of this matrix is a little bit nasty and its inverse is much easier to deal with. Can I claim ...
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1answer
15 views

Optimization, multivaraible

I have a question regarding multivariable optimization. In particular, I have a function f(x,y,z,w) and I want to maximize f in terms of x only (with other variables treated as parameters). Also I ...
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1answer
21 views

Can Gomory's cutting plane be used to solve Mixed Integer Linear Programs?

Do you know if Gomory's cut can be used to solve MILP problems ? I have read that Gomory's cut is useful when all variables are integer, but what if just some of them are integer ? Is is necessary ...
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Nearest and farthest point from a function to another [on hold]

Find the nearest and farthest point from the ellipse $ x^2 + 3y^2 =3 $ to the segment made by $ x+y = 3 $ in the first quadrant. Found in a multivariable calculus course. So I have to find the ...
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1answer
24 views

How to find the smallest value by using Lagrange multiplicators?

Let $a$, $b$ and $c$ be positive constants. How one can find the smallest value of the sum of three numbers $x_1$, $x_2$ and $x_3$ at the surface $\dfrac{a}{x_1}+\frac{b}{x_2}+\frac{c}{x_3}=1$ by ...
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19 views

Homography between known and unknown rectangle corners

I would like to know if there is a solution for the problem of homography estimation in the special case in which one of the views is unknown but has some constraints, particularly if we know the ...
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15 views

Closest Positive-Definite Matrix Subject to a Contraint

Given a positive, semidefinite, real 2n by 2n matrix $A$, is there a formula or an algorithm that finds the closest (in some sense, preferably Frobenius distance) positive, semidefinite, real 2n by 2n ...
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24 views

Gradient descent in inequality constrained optimization problems

I want to solve an optimization problem using a gradient descent algorithm maximize $$ max \log( \frac{Ax + b}{ Cx + b} ) $$ $$s. t. \quad 0 \le x \le 1 $$ where x is a vector and the ...
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38 views

Need help setting up an optimization problem. [on hold]

As part of the balancing efforts for the real time strategy game we are developing,we want you to investigate a particular gaming scenario. The ground units are constructed at the Vehicle Assembly ...
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1answer
25 views

Find a maximum triangle that lies on a polyline (with constraints)

If there's a polyline (a GPS track, actually) with a lot of points (could be several thousand), that looks like this 1) How can I find such a triangle with the biggest possible perimeter, that its ...
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33 views

By looking at a graph of a function, how do I find the maxima/minima of its curvature function. [on hold]

All Ideas are appreciated. I could think of some intuitive ideas, but could not back them by solid clean reasoning. thanks, I will post If i find anything
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143 views

How find the maximum of the $x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}$

Let $$0\le x_{i}\le i,\, i=1,2,3$$ be real numbers. Find the maximum of the expression $$x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}$$ My idea: I guess $$x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}\le ...
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3answers
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Solving a optimization problem

Here is the objective function of my optimization problem: $$ \min \left( \sum_{i=1}^{n}a_i(1 - X_i)\right), \qquad n = \arg \min(X_i = 1) $$ $$X_i = \{0,1\} \text{some other linear constraints} $$ ...
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37 views

Global optimization

Assume that I want to find the global minimum of a non-linear, non-convex, multidimensional function subject to several restrictions. Could you recommend me any deterministic strategy which can ...
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2answers
81 views

What numerical methods are known to solve $L_1$ regularized quadratic programming problems?

What numerical methods are suitable to solve the following problem $$\min_x \tfrac{1}{2}x^T A x + b^Tx + \lambda ||x||_1$$ where $x,b\in\mathbf{R}^n$, and $A\in \mathbf{R}^{n\times n}$ is positive ...
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Application of a general “Weierstrass theorem”

http://books.google.at/books?id=9OSrV73a40gC&pg=PA45&lpg=PA45 gives a general Weierstrass theorem. Are there notable applications of this theorem, say in the calculus of variations? (I could ...
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38 views

revenue optimization under multinomial logit

Let $[N]= \{1,...,N\}$ denote a set of items, item $i$ has an utility equal to $u_i > 0$ and a unit revenue of $r_i >0 $. Without loss of generality, assume that $$r_1 \geq r_2 \geq ... \geq ...
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formulate optimization problem in MATLAB [closed]

I have set of weights: lets say my long only portfolio looks like this id weight bweight GROUP A 0.25 0.3 T B 0.1 0.25 T C 0.05 0.25 E D 0.6 0.2 E I have a 4x4 cov as well I would like to add ...
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1answer
41 views

Find maximum of the function

I have the following target function $$ f(m,q)=\sum^{N}_{i=1}|m_i-q_i| $$, where $$m,q\in R^N$$ and $$\sum^{N}_{i=1}m_i=1, \forall m_i>0$$ $$\sum^{N}_{i=1}q_i=1, \forall q_i>0$$ I would like ...
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1answer
33 views

Strong convexity of a function with cases

Given a set $S = \{x_1,\dotsc,x_n\} \subset \mathbb{R}$, is the function \begin{align} f&: (0,\infty) \to \mathbb{R} \\ f&(p) = 2p^2 + \frac{1}{n}\sum_{i=1}^n \max(0, -p^2-x_i) \end{align} ...
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23 views

dual value of a linear constraint

Assume a minimization problem. The dual of an inequality '<' constraint is the marginal improvement in the objective function (ie marginal reduction) by marginally increasing the right-hand-side ...
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2answers
35 views

Finding extremal values on a set

Let $f(x,y)=(x-1)^2+y^2+xy$. Find the maximal and minimal values of $f$ on the set $M=\{(x,y):|x|+|y|\leq4\}$. Attempt: By taking partial derivatives and solving the homogenous algebraic system we ...
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2answers
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minimization with many unknowns and one condition

I haven't done this in quite a while so excuse my perhaps silly question. I'm looking for a solution to a minimization problem (if there is one), that goes like this: I want to minimize (global) ...
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39 views

Question about the ellipsoid method

I have some technical question concerning the ellipsoid method Referring to the paper : http://paswkshop.comm.utoronto.ca/~weiyu/01658226.pdf It is mentioned in p.1317 at the last line in the left ...
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1answer
32 views

Extreme points in compact convex domain [closed]

Let $f(x)$ be a continuous quasi-convex function in $R^n$ and let be a compact convex domain. If one denotes $ \Gamma:=[x\in \Omega: \operatorname{arg max}_x f(x) ]$ then $ \Gamma$ contains an ...
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24 views

Finding optimal hyperplane

I have a set of vectors $\{V_i\}$ in $n$-dimensional space. There is a number corresponded to each vector $\alpha_i = f(V_i)$ ($\alpha_i$ can be negative). I want to find a hyperplane which would ...
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1answer
39 views

Matrix optimization

I'm trying to minimize over $U$ the objective $\|X^{\top}UU^{\top}UU^{\top}X\|_F^2 = \text{trace}(X^{\top}UU^{\top}UU^{\top}XX^{\top}UU^{\top}UU^{\top}X)$ subject to $U^{\top}U = I$, where $X \in ...
2
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2answers
51 views

Lagrangian method gives the wrong answer in a standard constrained optimization problem

I had this strange problem where the Lagrangian method gives the wrong answer in a constrained optimization problem. Here goes: The problem is $$\max_{c,n,q} \alpha\log(c)+(1-\alpha)\log(nq)$$ ...
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1answer
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Is $argmin_{\mathbf{x}} f(\mathbf{x})=argmin_{\mathbf{x}} \log{f(\mathbf{x})}$ always true?

Assuming $\mathbf{x}\in \mathbb{R}^n$, $f(\mathbf{x})\gt0 \forall\mathbf{x}\in\mathbb{R}^n$, is $argmin_{\mathbf{x}} f(\mathbf{x})=argmin_{\mathbf{x}} \log{f(\mathbf{x})}$ always true? Why?
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80 views

Minimum of $|\det(X+iC)|$

Let $C$ be a fixed real $n\times n$ matrix, $X$ be an arbitrary real $n\times n$ matrix. Find the minimum value of: $$|\det(X+iC)|=\sqrt{\det(X+iC)\det(X-iC)}$$ When $n=1$ it's clear that the ...
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47 views

How to determine if a convex polytope is contained in a union of convex polytopes?

Given that we are in a Euclidean space of dimension d, that we have a bounded convex H-defined polytope P, and N possibly unbounded convex H-defined polytopes, I am looking for an "efficient" ...