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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Maximization problem doesn't work [on hold]

I have a maximization problem which I've modeled it with YALMIP. but there is a big problem, I don't know why its variable takes lower bound and doesn't try to maximize objective function!! could any ...
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31 views

Strategies to work with system of trigonometric inequality

I'm trying solve this problem using matlab, anybody know good strategies to work with system of trigonometric inequalities such as $ ...
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17 views

sum of logarithms of linear-fractional functions Optimization Problem

I am new to optimization theory and I am facing this optimization problem. \begin{equation} maximize \qquad f(x) = \sum_{i} ...
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22 views

multivarable optimization problem, what is the procedure?

Sorry for this obvious question. I am trying to maximize an objective function that consist of 5 variables (a,b,c,d,e) over a and b. That is , $max _{a,b}f(a,b,c,d,e).$ So I procedure I took is ...
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16 views

Minimization with two functions that are not completely related

Two caveats: 1) This is a problem I formulated myself, and so may not be structured correctly/logically. 2) I don't have an extensive math background, but am currently finishing up Calc 3. I have an ...
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34 views

Proving boundedness of a function.

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
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79 views

Unsolvable(?) Assignment Problem

I've recently been trying to implement the Hungarian Method in C++, and I've been using 5x5 matrices to test my program. Last night I came across a matrix which neither I nor my program can solve. Is ...
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1answer
21 views

Volume of a polytope cut off by a hyperplane

Given a maximization problem with constraints, and adding a few more constraints using the Gomory cuts and solving the relaxed maximization problem, we can arrive at integer solutions. I am looking to ...
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18 views

Dimensional Consistency in Grids used in Optimization

I am working on an optimization problem in the research I am doing and my partner and I have found that in order to quickly converge on a solution using a specific PSO (the firefly algorithm - it's ...
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18 views

Find unknown such that four dependent quantities have the same value.

I have $12$ unknown $a_i, b_i, c_i, i=1,\ldots,4$, that should satisfy equations $$ \sum_{i=1}^4n_ia_i=a,\quad\sum_{i=1}^4n_ib_i=b,\quad\sum_{i=1}^4n_ic_i=c, $$ where $n_i,\,i=1,\ldots,4$ and $a,b,c$ ...
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Strong duality: When does the optimal primal variable coincide with the primal variable giving the dual function.

I'm considering the inequality-constrained optimization problem of finding $$ x^{\star} = \arg \min_{x} f(x) \;\; \text{s.t.} \;\; h(x) \le 0 $$ which is assumed to have a unique minimizer. The ...
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36 views

Given a satisfactory real number = [any integer]/(2b) where a and b are integers, how would one find the minimum value of b?

For instance, 0.625 = 5/(2*4). Given 0.625, how would one find 4? 0.75 = 1/(2*2). Given 0.75, how would one find 2? I should ...
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26 views

compute a certain maximum in MATLAB

let $ C \in \mathbb{N} $ and $ c_1>c_2>\ldots>c_k \in \mathbb{N} $ with $ C>c_1 $ and $ c=(c_1,c_2,\ldots,c_k)^\top \in \mathbb{N}^k $, where $ \mathbb{N} $ are the natural numbers without ...
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46 views

Find scaling factor that minimizes f(x) - round(f(x))?

Let's say I have a function $f(x)$, which has a fractional component $\{ f(x) \} = f(x) - \lfloor f(x) \rfloor$. I would like to add a scaling factor $h(x)$, where $h(x)$ is a polynomial, such that ...
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16 views

Hyper-plane that separating hyper-cube.

Suppose $\Omega \in \mathbb{R^4}$ is closed unit ball in $ ||.||_{inf}$ i.e. Hyper-cube. 1) Am I right that there are L=16 extreme points of $\Omega$, all are vertices of the hyper-cube. 2)Is it ...
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2answers
33 views

A curious question about optimizing a function of 2 variables.

Let $f(x,y)$ be defined and has continuous first and second partials on a domain $D$. Also, let $$A = \frac{\partial^2 f}{\partial x^2} \\ B = \frac{\partial^2{f}}{\partial x \partial y} \\ C = ...
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2answers
26 views

How to introduce flat cost of flow over a node using mixed integer programming.

In the set up for the program we have a graph where we are trying to minimize the cost of sending flow over the arcs. I have formulated the following linear program. \begin{array}{ll} \text{minimize} ...
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4answers
24 views

Help with Lagrangian Constrained Optimisation

Question: Maximise f (x, y) = x2y, where (x, y) ∈ R2 given the constraint that all (x, y) are points on a circle with radius √3 around origin (0, 0). Solution: f (±√2, 1) = 2 is the maximal value ...
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23 views

Unconstrained optimization problem (lasso with modification)

I am looking to solve the following unconstrained optimization problem: $$\arg \min_U \|b-A(UY^*)\|_F^2+\lambda\|U\|_1$$ where $\|.\|_F$ is frobenius norm. I know that the solution without the ...
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1answer
19 views

Optimum set partitioning with constraint

Be $A \subset D \wedge m \in D \wedge \forall x \in A:x < m$, with $D$ finite and included in the positive integers, I need to partition $A$ into $B_n$, while minimizing $n$, so that ...
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2answers
29 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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1answer
29 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
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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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2answers
29 views

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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1answer
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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1answer
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
28 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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23 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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1answer
29 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
27 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
37 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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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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1answer
49 views

linear programming, product mix [on hold]

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 ...
2
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0answers
42 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
29 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
24 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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1answer
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Nearest and farthest point from a function to another [closed]

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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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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1answer
19 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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31 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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0answers
38 views

Need help setting up an optimization problem. [closed]

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 ...
3
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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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1answer
35 views

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

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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4answers
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 ...