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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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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35 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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Is this permutation-sum problem NP-hard?

A new, tighter tardiness bound has been found for global Earliest-Deadline-First scheduling of jobs on symmetric multiprocessors. But this bound seems to be particularly hard to compute. In ...
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7 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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2answers
27 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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29 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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12 views

Gradient descent in Banach space

Can somebody explain me why gradient descent in Banach space does not make sense? As pointed out by Sebastien Bubek (https://blogs.princeton.edu/imabandit/2013/04/16/orf523-mirror-descent-part-iii/) ...
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10 views

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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1answer
17 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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1answer
61 views

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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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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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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3answers
319 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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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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450 views

Can a dynamic programming problem be transformed into a linear algebra problem?

Here is a simple standard economic problem: Let Robin Crusoe have an endowment $w_0$ of lembas bread. She is immortal and discounts at a rate of $\beta$ per period. Each period (from $t=0$ on) she ...
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27 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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11 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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2k views

primal to dual solution conversion ??

i have an optimization problem $$\text{ maximize } z=3x+4y$$ $$\text{ such that: } x+y ≤ 450 \text{ and } 2x+y ≤ 600$$ the optimal solution to this problems comes to be $x=0$; $y=450$; $p=150$ ...
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9 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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1answer
501 views

Maximum likelihood estimators, hypergeometric and binomial

I'm trying to solve a two part problem. The set up is as follows: consider a bag with $\theta$ red marbles and $7-\theta$ blue marbles, with $\theta$ being unknown. Let $x$ denote the number of red ...
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19 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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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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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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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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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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1answer
33 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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3answers
150 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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12 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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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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305 views

The longest sequence of numbers with a certain divisibility property

Definition - Denizen A sequence $\lbrace a_k \rbrace$ is a denizen if all of it's members are prime numbers, i.e $a_0, a_1, ... a_n \in \mathbb{P} $; and it satisfies the following condition; ...
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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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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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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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28 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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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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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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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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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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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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Minima problem?

This is a question in my textbook which I can't solve. Any help would be appreciated, thanks. "A piece of wire 10 metres long is cut into two portions. One piece is bent to form a circle, and the ...
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5k views

What does “curly (curved) less than” sign $\succcurlyeq$ mean?

I am reading "Convex Optimization" by Stephen Boyd. He is using a curved greater than and curved less than equal to signs. $f(x^*) \succcurlyeq \alpha$ or $f(x*) \preccurlyeq \alpha$ Can someone ...
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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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41 views

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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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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1answer
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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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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33 views

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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291 views

Find pressure in a sinusoidal function

Tiffany is a model rocket enthusiast. She has been working on a pressurized rocket filled with laughing gas. According to her design, if the atmospheric pressure exerted on the rocket is less than 10 ...
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3answers
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 ...
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1answer
26 views

Maximizing a convex function under constraints

Consider the following non-convex problem: \begin{equation*} \begin{aligned} & \text{maximize} & & f(X) \\ & \text{subject to} & & f(X)\le b\\ &&& A_kX = c_k, \ ...