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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Functional optimization: maximize a double integral where the functional appears twice

Please help me solve the following optimization problem. Suppose that you have to choose a function $U: [0,1]\mapsto [0,1],$ which must be nondecreasing ($U'\geq 0$) to maximize the following ...
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10 views

Show that in Line Search Methods the “steepest descent direction” is the one along which the objective function decreases most rapidly

I want to verify the claim, that the steepest descent direction $-\nabla f(x^k)$ is the one along which $f\in C^2(\mathbb R^n)$ decreases most rapidly. Therefore, I considered the Taylor expansion ...
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Find local extrema of $x_1+5x_2$ when $2x_1+3x_2=1$ and $x_1-x_2+x_3=0$.

I'm trying to solve the following problem: Find all local extrema of $f:M\to\mathbb{R}$ where $$ M :=\left\{x\in\mathbb{R}^3: 2x_1+3x_2=1,\ x_1-x_2+x_3=0\right\} $$ and $$ f(x)=x_1+5x_2,\ ...
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20 views

Minimize the volume bounded by a plane

A plane passes through the point $(a, b, c)$. Find its intercepts with the coordinate axes if the volume of solid bounded by the plane and the coordinates planes is to be a minimum. What I have ...
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Critical Points of lagrangian in Kuhn-Tucker

I am reading Sundaram's "a first course in optimization theory". In section 6.2.2, he is explaining why K-T conditions works. And he describes those $(x,\lambda)$ which satisfies K-T conditions as ...
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Minimum value of $\cos x+\cos y+\cos(x-y)$

What is the minimum value of $$ \cos x+\cos y+\cos(x-y). $$ Here $x,y$ are arbitrary real numbers. Mathematica gives (with NMinimize) $-3/2$. But I don't know if this is correct and if so, how to ...
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If $A^k$ consistently approximates $\nabla^2f(x^k)$ with $x^k\to x^*$ and $\nabla^2f(x^*)$ regular, then the $A^k$ are regular, too

Let's call $\left\{A^k\right\}\subseteq\mathbb R^{n\times n}$ a consistent approximation of $\left\{B^k\right\}\subseteq\mathbb R^{n\times n}$ iff ...
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23 views

Probability or optimization

I have a problem with the following case $F$ and $G$ are distribution function on $x\in{[0,1]}$ and they have same mean $\mu$ I want to prove $\int_0^1 F(x)G(x)dx\geq(\mu-1)^2$
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17 views

optimize matrix equation (low rank)

I am trying to solve the following optimization problem $min_{H,V}$ $||A-HV||_F^2$ s.t $V\geq 0$ (i.e all entries are non-negative) and H is low rank Is there a way to tackle the is problem Can ...
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optimization query

i am trying to optimize the following matrix function $\min_{H,V}$ $\|A-HV\|_F^2$ under the following constraints (1) $H$ is low rank (2) $V$ is positive (all entries of $V$ are non-negative) Any ...
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30 views

Measuring the effect of a linear transformation on the result

I have an unknown vector $x\in\mathbb{R}^n$, a known orthogonal matrix $\Phi\in\mathbb{R}^{n\times n}$, a known matrix $A\in \mathbb{R}^{m\times n} (m \le n)$, and a known vector $b\in \mathbb{R}^m$ ...
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6 views

Why do we transform constrained optimization problems to unconstrained ones?

In methods like Lagrange multipliers or augmented Lagrangian methods we transform a constrained optimization problem into an unconstrained one and then solve it. For example in Lagrange multipliers ...
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58 views

Conditions on $c$ such that the inequality dont hold.

I want to find conditions on $c$ such that the inequality don't hold. $$1-ac(a-2)(a-1)^2 < 0 \ \ \ \ \ \ \text{for } a>2, c>0$$ If $\phi(a) = ac(a-2)(a-1)^2 \Rightarrow \phi'(a) = ...
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16 views

Global optimization methods where constraints are lipschitz functions

Is there any global optimization methods where objective function is nonlinear (not lipschitz) but constraints are lipschitz functions?
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57 views

Optimizing sums of log det

I have a set of points $S$ which have to be clustered into $K$ cluster say, $S_k$, by minimizing the following function: $J = - \sum_{i=1}^{K} \log \det( \mathbf{I} + H_i H_i^T)$, Where $H_i$ is the ...
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The gradient of a function on a Banach space is an element of the dual space

Can somebody explain me why gradient descent in Banach space does not make sense? As pointed out by Sebastien Bubek in his blog, the gradient is an element of the dual space $\mathcal{B}^*$. But I ...
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I have finite resources, with 'n' number of items at different prices

Suppose I have finite resources, say 500 dollars. I have the choice of purchasing 'n' items, and choosing between 'i' different items, say i=5. Each item is priced differently, in this case, let's say ...
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2answers
30 views

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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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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19 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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28 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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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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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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Intuitive meaning of “Primal Dual Interior Point Method” [closed]

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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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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Big $O$ question for While and For loops [closed]

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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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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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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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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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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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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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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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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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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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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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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36 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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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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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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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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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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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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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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38 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 ...