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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Approximate solution to a matrix equation

Let $A$ and $B$ be $n \times m$ matrices. I am looking for a $m \times m$ matrix $X$ which would be an approximate solution to the equation $AX = B$ (an exact solution is very unlikely to exist). More ...
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13 views

Find the distance such that the angle will be the gratest

Rectangle shaped screen in a cinema is 8m high. It is place on a wall in such a manner that the upper edge of the screen is 12m above the floor. Find the distance between the viewer and the wall where ...
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50 views

Why in general there is no systematic way to find counterexamples? What kind of property do they all break that lead to this? and other things

We came across counterexamples in many areas of mathematics: For example Sum of irrational numbers not necessary being irrational The "Windmill blade" function (for lack of a better name of one of ...
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Find minimum distance between the plane and the beginning of Cartesian plane.

Find minimum distance between the plane: $S=\{\left(x,y,z\right) \in \mathbb{R}^3: x+yz=2012 \}$ and the beginning of Cartesian plane $(0,0,0)$. I want to minimize this with use of lagrange's ...
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8 views

About finding the common diagonalizing similarity transformation.

Say I have $2k$ matrices $M_{a_1b_1}$, $M_{a_2b_2}$,..,$M_{a_kb_k}$ and their negatives. Here $M_{a_ib_i}$ is such that it has $0$ everywhere except that it has $1$ at $(a_i,b_i)$ and $(b_i,a_i)$ ...
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23 views

Minimum of sum of squares over sums

I am trying to minimize $\phi(\alpha)$, where $\alpha \in \mathbb{R}^K$. $\phi(\alpha) = \frac{R^2 + G^2 \gamma \sum_{i=0}^{K} A_i \alpha_i^2}{\sum_{i=0}^{K} A_i \alpha_i} $ Where, $A_i = \gamma ...
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7 views

Can variance be replaced by absolute value in this objective function?

Initially I modeled my objective function as follows: argmin var(f(x),g(x))+var(c(x),d(x)) where f,g,c,d are linear functions in order to be able to use mixed integer linear solvers, I modeled the ...
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8 views

Question on applications using schur complements

i wonder if you may be able to contribute some areas/ideas where the use of schur complements are used. Like for exampple, I think schur complements can be used to check for positive definiteness of ...
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18 views

Intuitive meaning of “Primal Dual Interior Point Method” [on hold]

I am trying to understand how "Primal Dual Interior Point Method" works for nonlinear optimization. I have seen some examples already. Wikipedia has a very good example too. But I am still finding it ...
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8 views

Is there a solid reference work that covers optimization for discrete and for continuous domains?

I am looking for a good, comprehensive reference on optimization. Currently, I have Lundberg's "Linear and Nonlinear Programming, 3rd Ed", but this completely omits integer programming, except in the ...
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19 views

Big $O$ question for While and For loops [on hold]

I have to find the exact $O(N)$ for these instructions, not just the order of magnitude. I'm not getting any of the answers provided for me. I know the first loop is $O(3N+2)$. The declaration of ...
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16 views

Optimization problem with distance multiples

Having a set of real positive distances {di} where i goes from 1 to N, the optimization problem is as follows. We want to find the set {di} such that no pair [di,dj] has a common multiple, and ...
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12 views

Optimization of shoe manufacturing

I cannot seem to figure out the best way to optimize the shoe manufacturing algorithm in order to minimize the costs in the company I work for. Let me describe the problem a bit. A customer makes ...
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11 views

Signal recovery using Majorization-Minimization with Quadratic Upper Bound

I am trying to formulate a majorization-minimization (MM) (via quadratic upper bound) approach to total variation denoising (TVD). The total variation denoisng objective function is defined as an ...
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139 views

Finding the maximum of a function on $ \Bbb{S}^{7} $.

I'm trying to find the maximum of the function $$2 a^2 h+\sqrt{3} a d f+\sqrt{3} a e g+2 b^2 h-\sqrt{3} b d g+\sqrt{3} b e f\\+2 c^2 h+\sqrt{3} c d^2+\sqrt{3} c e^2-\sqrt{3} c f^2-\sqrt{3} c ...
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24 views

How to calculate optimal sizes of rectangles for this type of array visualization?

Given array of positive numbers, I would like to draw this diagram and be able to put descriptions inside: There should be no empty space left, consider that these numbers represent % of total. Do ...
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2answers
24 views

Underdetermined Equation Optimization

For the equation: $$5X + Y + Z = 600$$ With constraints: $$92 \le X \le 95$$ $$46 \le Y \le 55$$ I want to find a method that will choose values for $X$ and $Y$ such that $\lvert Z\rvert$ is ...
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8 views

How to compute the gradient of the weighted kernel

Let's say $f(X) = \sum_{i,j}A_{i,j}x_i'\cdot x_j $ where $x_i,x_j$ are the i-th, j-th columns of $X$. So what is the gradient $\frac{\partial(f(X))}{\partial{X}}$ ?
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16 views

Constrained optimization with several equality constraints

In maximizing a function of $n$ variables with $m$ equality constraints, it is required that the Jacobian derivative of constraints has full rank at optimal points. Can some one provide me with the ...
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27 views

Lagrange multiplier vs KKT

Suppose task 1: maximize $f(x, y)$ subject to $g(x, y) = 0$ and $h(x,y) = 0$ Suppose task 2: maximize $f(x, y)$ subject to $g(x, y) \geqslant 0$ and $h(x,y) = 0$ According to wiki for the first ...
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53 views

Why is the Lagrange Multipliers Theorem not working?

Consider the function $h: K \to \mathbb{R}$ where $K := \{x \in \mathbb{R}^3:x,y,z \geq 0, x+2y+3z\leq 6\}$. $h$ is defined as: $$ h(x) = xe^{(x+2y+3z)} $$ Find the supremum and the ...
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1answer
31 views

Warm start of simplex algorithm after update of constraint matrix

Assume we found an optimal solution $\mathbf{x}_1$ of the linear program \begin{gather} \max \mathbf{n}^T\mathbf{x}\mbox{ s.t. }A\mathbf{x} \leq \mathbf{b}\tag{1} \end{gather} using the simplex ...
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1answer
15 views

Find maxima and minima of the function

Given: $$f:\mathbb{R}^2 \rightarrow \mathbb{R}, f\left(x,y \right)=-x^4+x^3-3x^2y+3xy^2-y^3$$ Find all points where gradient is equal to zero. Decide whether in those points function has either maxima ...
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6 views

concavity conditions with respect to s for $\Pi \left( s \right)=\underset{x}{\mathop{\max }}\,f\left( s,x(s) \right)$

Here is the function: $\Pi \left( s \right)=\underset{x}{\mathop{\max }}\,f\left( s,x(s) \right)$ I want to find the conditions of showing $\Pi \left( s \right)$ is concave with respect to s at $x^*$ ...
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29 views

Algorithm for maximizing the overlap between sets of voxel points

I have a problem that I've formulated as follows. Given a finite target set $T$, and a set-generating function $F(x_i) = C_i$ that also produces finite sets, I'd like to find the set $C_i$ that has ...
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24 views

Submodular function, square of which is also submodular?

A Submodular function $ f:2^E \rightarrow R $ is a function that satisfies the following two equivalent definitions: for every $ S,T\subseteq E: f(S) + f(T) \geq f(S\cup T)+f(S\cap T) $ for every $ ...
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34 views

convex optimization?

I have a question about the convexity of an optimization problem and its solution. Suppose $f(X)=-tr(A^{T}XA)+tr(X)$, $A$ is any matrix with its dimension "matched" with $X$. The optimization problem ...
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29 views

Local global minimizers and maximizers

I want to find the local and global minimizers and maximizers of the following two functions. 1) $f(x)=x^2e^{-x^2}$ 2) $f(x)=x+ \sin x $ These are my answers. 1) $f(x)=x^2e^{-x^2}$ ...
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Find the maximum and minimum of $\sum \limits_{i=1}^n x_i ^3$

Let $x_1,x_2, \dots ,x_n$ be a sequence of integers such that $i) -1\le x_i\le 2$ for $i=1,2,\dots,n$ $ii)x_1+x_2+\dots+x_n=19$ $iii){x_1}^2+{x_2}^2+\dots +{x_n}^2=99$ Determine the minimum and ...
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27 views

Maximum volume inscribed ellipsoid inside nonconvex polyhedron

In Convex Optimization (Boyd, Vandenberghe), an algorithm for finding the maximum volume inscribed ellipsoid inside a convex polyhedron is given on p. 414 (8.4.2 Maximum volume inscribed ellipsoid) [I ...
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The value of $a$ for which $f(x)=x^3+3(a-7)x^2+3(a^2-9)x-1$ have a positive point of maximum lies in the interval $(a_1,a_2)\cup(a_3,a_4)$

The value of $a$ for which $f(x)=x^3+3(a-7)x^2+3(a^2-9)x-1$ have a positive point of maximum lies in the interval $(a_1,a_2)\cup(a_3,a_4)$.find the value of $a_2+11a_3+70a_4$ I differentiated ...
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38 views

Find the cosine of the angle at the vertex of an isosceles triangle having the greatest area

Find the cosine of the angle at the vertex of an isosceles triangle having the greatest area for the given constant length $l$ of the median drawn to its lateral side. I tried to solve this ...
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3answers
37 views

An upper bound for a function

I am trying to find an upper bound $b\ge f(x)~\forall x\ge0$ for the following function: $$f(x)=\frac{x}{(w+ux^2)^2},$$ where $w,u>0$ are parameter values. I am interested in the positive domain ...
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Increasing a singular value [closed]

Can any one tell me the effect of increasing one singular value (say 10 times ) larger than others.Whether it has any importance in optimization Problems .
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23 views

How to prove $\int_{\Omega} \sum_{i=1}^{N} f_i(x)dx$ is equivilant with $\sum_{i=1}^{N} \int_{\Omega} f_i(x)u_i(x)dx$

I have a 2D image in $\Omega$ space. Assume that the space can be separated into $N$ sub-regions $\Omega_i$ such that $\Omega_i \cap\Omega_j=\emptyset$; $\Omega_i \cup \Omega_j=\Omega, \forall ...
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3answers
24 views

Global Optimization of a well-defined function with gradient information

I try to minimize the function $$ f(x_1, …x_n)=\sum\limits_{i}^n-a_icos(4(x_i-b_i)) +\sum\limits_{ij}^{edge}- cos(4(x_i-x_j)) $$ $$x_i,b_i\in (-\pi, \pi)$$ where $\sum\limits_{ij}^{edge}$ only sums ...
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35 views

minimize smallest eigenvalue

Assume $P_A,P_B$ are probability transition matrices (each element is nonnegative and row sum is 1) and $v$ is probability row vector (each element is nonnegative and sum of elements is 1). How to ...
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34 views

Proving Convergence of a Derivative Free Method

Hopefully this question is on topic for the mathematics community (rather than the statistics) since the optimization method I am using relies on a statistical model. Well I am building an algorithm ...
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1answer
24 views

Condition for guaranteed minimum-rank solution

Consider the following rank minimization problem of a positive semi-definite matrix $X$: \begin{equation*} \begin{aligned} & \underset{X}{\text{minimize }} & & rank(X) \\ ...
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simplify/solve nonlinear equations for constrained least squares problem

I am trying to find a simple, ideally closed form formula for the (not necessarily unique) unit vector $\vec{x}$ minimizing total squared cosine distance from a collection of unit vectors $\vec{v_i}$. ...
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2answers
57 views

How to find extrema of $\sqrt{x_1^2 + x^2_2 + x^2_3}$ defined on $\{x \in \mathbb{R}^3 : x_1^2 + 2x^2_2 + 3x^2_3 < 1\}$

I have a function $g: U \to\mathbb{R}$ where $$U :=\{x \in \mathbb{R}^3 : x_1^2 + 2x^2_2 + 3x^2_3 < 1\}$$ and $$g(x) = \sqrt{x_1^2 + x^2_2 + x^2_3}$$ I would like to find out if g(x) has any ...
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1answer
35 views

How to minimise the upper boundary of this weird function?

Let $\{x\}$ denote the fractional part of $x$, which is $\{x\}=x-[x]$. Let $f_{a,b}(x)=\{x+a\}+2\{x+b\}$ and let its range be $\{m_{a,b},M_{a,b})$. Find the minimum value of $M_{a,b}$ as $a$ and ...
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1answer
39 views

How can I find the circumference of a circle using optimization? [closed]

I need Help ASAP!!!! I have a circular gutter which has a sections of the circle taken out. the total area of the circle is 8600mm^2 and i don't know how to get find the perimeter or circumference of ...
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1answer
16 views

how to interpret local minima of combinatorial optimization

I am having a difficult time trying to interpret and visualize the local minima of a combinatorial optimization objective function. Here's a rough sketch of my problem: I have $m$ points ...
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33 views

How to minimize the following functional? [closed]

I want to minimize following energy. \begin{align*} E(f,h) &= \int_\Omega Ah + Bf dx \\ \text{subject to }& \lVert f \rVert=1 \end{align*} where $A,B$ is a constant $m\times n$ matrix, ...
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1answer
33 views

Graphical solution (with two variables), solution properties.

(c) infeasibility depends on the constraints; if we look at the graph we can see that (1, 1) is the intersection of constraint (I) and (II), and for this to be infeasible we need t in constraint ...
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7 views

How to optimize a system of equality an $\geqslant $ constraints?

In many cases, for example when we work with probably mass functions We may need to solve a system of this form: $$ max f(\vec{p_1})+g(\vec{p_2}) $$ when there are the obvious constraints of : $$ ...
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2answers
111 views

Are these two optimization problems equivalent to each other?

Let $\mathbf{x}=[x_1,\ldots,x_K]^T$. For a fixed vector $\mathbf{a}$, I have the following optimization problem : \begin{array}{rl} \min \limits_{\mathbf{x}} & | \mathbf{a}^T \mathbf{x} | \\ ...
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0answers
28 views

Reason for use $L^2$-Norm instead of $L^1$-Norm in Optimization [closed]

In optimization we use $\min\; \Vert Ax-b\Vert_{2}^2$ instead of $\min\; \Vert Ax-b\Vert_{2}$ because second is not differentiable. But I am looking for a clean and mathematical reason for this. And ...
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1answer
16 views

Inspecting a project network

From the given table one can draw the project/activity network above. There are four possible paths: (i) C, B. (ii) C, A, D. (iii) E, D. (iv) G, F, D. The first path's project time is 4+5=9, the ...