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.

learn more… | top users | synonyms (5)

0
votes
1answer
37 views

Find those values 'a' which belongs to the Convex Hull

Find those values of 'a' for which (1,a,1) belongs to the convex hull of $$\{(0,0,0), (1,1,2),(2,4,-6), (1,3,8)\}$$ Give me hints as much as you can, I would like to understand the mindset rather ...
0
votes
0answers
31 views

Solving this in order to L: (6x-L)(6y-L)-9=0

I don't really know how to explain this in english since I study it in portuguese, but I can't get my head around to solve this. My book says the solution to this problem is L1=6x V L2=36x^2 * y^2 -9 ...
0
votes
2answers
547 views

Optimal Basic Feasible Solutions

In linear programming, is it true that you can only have at most 2 optimal basic feasible solutions? If so, why?
1
vote
0answers
12 views

Optimizing a set of rules to better predict the outcome of events

I'm trying to better predict the top three finishers of the next 1000 800m mens freestyle swimming race. I've got a set of rules to rate the swimmers: 1) Add 5 points if the swimmer won his last ...
0
votes
1answer
21 views

One solution of a diophantine system

How to find one solution of $Ax = b$, where $A$ is a $(m, n)$ matrix and $x$ a vector of size $(n, 1)$. $A$, $x$ and $b$ are matrices of integers entries. How to check whether is a solution exists?
0
votes
1answer
32 views

Conditions for unique solution of a maximization problem?

Let $S\subseteq \mathbb{R}^2$, $d:=(d_1,d_2) \in S$, and $s:=(s_1,s_2)$ a generic point of $S$. Assume that there exists $s \in S$ such that $s_1>d_1$ and $s_2 >d_2$. Consider the following ...
0
votes
0answers
4 views

A Special Case of Maximum Coverage Problem

Let us denote $[N]=\{1,2,\cdots,\ N\}$ and let $\{a_{ij}\}_{i,j\in [N]}$ are positive numbers. Then how to proceed to solve the following problem $$\max_{G\subset[N]:|G|\le K}\left(\sum_{i\in ...
1
vote
0answers
382 views

Moore–Penrose pseudoinverse reference

Given the eigendecompositions $AA^{\top}=Q \Lambda Q^{\top}$ and $A^{\top}A=P \Lambda P^{\top}$, where $\Lambda$ is a diagonal matrix (of eigenvalues) and $P$ and $Q$ are unitary eigenvectors matrices ...
0
votes
0answers
12 views

Question about norm of diference of minimum of functions

let's say you have a function $f(x,y):\mathbb{R}^n \times \mathbb{R}^n \mapsto \mathbb{R}$ which is differntiable with respect to $x$ and has (at least) a minimizer on that variable. Then define ...
0
votes
1answer
19 views

Solving for stationary points for questions of the following type

How do you solve questions like $f(x,y) = x^2y + y^3x -xy$ for stationary points? A link to an educational resource that goes over this would be very helpful as well, as I don't even know what ...
0
votes
1answer
19 views

Are there two notions of flow?

I'm reading Jungnickel's Graphs, Networks and Algorithms. He defines the flow as a mapping $f:E\to \mathbb{R}_0^+$, which seems to mean the value of the flow of each edge, but in here: When he ...
0
votes
0answers
19 views

Is there a cost function for row equivalent matrices?

I am working on a minimization problem as follows: argmin$_x$ ||x-y||$_2$$^2$+$\lambda$||$\Psi$x||$_1$ where x and y are 2D or 3D complex arrays ||$\cdot$||$_1$ and ||$\cdot$||$_2$ are the L1 and L2 ...
2
votes
2answers
29 views

Can calculus optimization problems be turned into linear programming problems?

I found a Linear Programming textbook somewhere, and I skimmed through the first few pages. While I am not nearly enough ready to go through it, the things it dealt with seemed very much like calculus ...
2
votes
1answer
26 views

Is there Lipschitz property for subdifferential?

I'm trying to bound the quantity $\langle \nabla \Psi(x),\bar{x}-x \rangle$ above, with the bound depending on $\|x-\bar{x}\|$ and perhaps also of $\|x-y\|$ for fixed (but not varying) points $y$. ...
2
votes
2answers
41 views

Maximizing $\log(|A|)-\text{Tr}(AB)$ for pd and symmetric $A$ and $B$

Let $A$ and $B$ be two symmetric and positive definite matrices of the same size. Then the function $$ f(A)\equiv\log(\det(A))-\text{Tr}(AB) $$ is maximized uniquely by $A=B^{-1}$. This is ...
1
vote
1answer
58 views

Preconditioning of optimization problems

This question suggests that you can precondition an optimization problem by a simple multiplicative scaling of the variables in the objective function. However, when I look up literature on ...
0
votes
0answers
13 views

How to work with difference-of-elements penalty in optimization

I am trying to solve the optimization problem $$\min_{H,S>0} \|W(H+S)-X\|^2_F+Q(H)+\eta\|S G\|_F^2$$ where $X\in\mathbf{R}_+^{m\times T}$, $W\in\mathbf{R}_+^{m\times k}$, ...
0
votes
0answers
18 views

Finding the Dual of a primal LP

Suppose that we have the following primal LP: $\min z=c^Tu+d^Tv \\ \mbox{s.t.}\ \ \ \ \ \ \ \ \ u+Av=b, u\geq 0, v\geq 0$ I want to find the dual problem of this LP but I am slightly confused as ...
0
votes
1answer
25 views

How to get the updating rules? after derivation

In the picture i brushed yellow, it dose make no sense to me to get formulas(2) and (3). If anyone could point out or give some references? Thanks a lot!
2
votes
2answers
64 views

Efficiency of a max-min problem for $\sum_{j=1}^m |b_j-a_j|$ with $a_i$, $b_j$ restricted to convex sets

Consider the following optimization problem: $$\max_{\{a=(a_1,a_2,\ldots,a_m)\in A\}}\min_{\{b:=(b_1,\ldots,b_m)\in B\}} \sum_{j=1}^m |b_j-a_j|.$$ Is computing the optimal value of this problem ...
3
votes
0answers
46 views

Finding L^1 centers of sets of probability distributions

Let $\mathcal{P}^n = \{ x \in \mathbb{R}^n : x \geq 0, \sum x = 1\}$. Suppose I have $p_1, \ldots, p_m \in \mathcal{P}^n$. I want to find an $L^1$ center for these points. i.e. $q \in \mathcal{P}^n$ ...
0
votes
1answer
41 views

applied optimization problem- triangle fence

A farmer is trying to fence off a field on the edge of a river. He has two 1km long sections of fence to use to make a triangular field. The edge by the river does not need fencing, and the fence ...
0
votes
0answers
15 views

Optimization problem-distance question

Jessica needs to get to a boat. the boat is 100m offshore. She is currently running along the beach 1km away from the closest point on the beach to the boat. She can run at 3m/s and swim at 1m/s. She ...
2
votes
1answer
25 views

What is the solution for this quadratic program?

Given scalars $p_1\geq p_2\geq \cdots \geq p_r > 0$, can we find a solution for following problem? \begin{align} \text{minimize} & & & \sum_{j=1}^{r} p_j (1-t_j)^2 \\ \text{s.t.} \\ ...
3
votes
1answer
464 views

Trace Minimization of Covariance Matrix

Given a matrix X whose rows contain observations collected at some locations. Can someone explain how trace minimization of covariance matrix $XX^T$ can lead to orthogonal / mutually independent ...
0
votes
0answers
14 views

Mathematical model for MILP optimisation problem- power scheduling

I am having trouble trying to formulate a mathematical model for the problem below to solve later as an MILP problem using different optimization software. I want to formulate the model so as to ...
0
votes
1answer
48 views

Prove max f(x) = -min -f(x)

How do I prove : $$max f(x)= - min -f(x)$$ I am trying to prove this, and have tried to use my book but I am stuck.
1
vote
1answer
27 views

Solving a Non-linear Multivariable System of equations

How would I go about solving a system of nonlinear equations where the highest degree is two? For example: $$f_1(x) = f_1(x_1, x_2,\dots, x_n) = 0,$$ $$f_2(x) = f_2(x_1, x_2,\dots, x_n) = 0,$$ ...
1
vote
1answer
68 views

Optimization of competitive scenario

Suppose we have a function $f(x_1,x_2)$ with the following properties: Let $x^*=\arg \max_{x_1} f(x_1,x_2=x^*)$ and $x^*=\arg \min_{x_2}f(x_1=x^*,x_2)$. $f(x_1,x_2)$ is concave in $x_1$. ...
0
votes
0answers
29 views

Using Euler-Lagrange to find the first variational curve of

I have been working on this optimization problem for days but I cannot figure out the right way to finish it off. I am reading from Optimization Theory by Pierre, and this is problem 3.3. Note that ...
-2
votes
1answer
23 views

How to find the maximum of $p=\sqrt{15x}+ \frac{x}{x-65}$ where $0< x <65$?

Given a demand function, $$p=\sqrt{15x}+ \frac{x}{x-65}$$ and $$0< x <65$$ Find the value of $x$ that produces a maximum value. I have no idea how to find the answer.
3
votes
4answers
72 views

Maximum and minimum of $f(x,y)=xy$ when $x^2 + y^2 + xy =1$

It is asked to find the maximum and minimum points of the function $$f(x,y)=xy$$ when $x^2 + y^2 + xy=1$ I've tried Lagrange and obtained $$\lambda = \frac{y}{2x+y}=\frac{x}{2y+x}$$ but what ...
4
votes
1answer
73 views

Smallest possible triangle to contain a square

I was looking at this stack exchange question* and started thinking about the case of a polygon with 4 sides: a square. The question asks for a program that can take a polygon of N sides and return ...
1
vote
1answer
22 views

Express this linear optimization problem subject to a circular disk as a semidefinite problem.

I have to express following problem as a semidefinite problem: $ min \, F(x,y) = x + y +1$ subject to (1) $(x,y) \in \mathbb{R}^2 : (x-1)^2+y^2\leq 1$ Only affin equality conditions should be used. ...
0
votes
2answers
42 views

Prove convexity of function over space of positive definite matrices

I want to show that the function $f(X) = -log \ det(X)$ is convex on the space $S$ of positive definite matrices. What I have done: It seems like this problem could be tackled by considering the ...
1
vote
1answer
36 views

Maximum remainder

I got this programming challenge - and can solve it by brute force. But I would love to solve it using a more mathematical approach. ...
0
votes
2answers
54 views

Optimization problem calculus 1

I always have trouble setting up the problem. I do not know what formulas I will need. I know volume is the constraint and surface area is what I have to find which have to be as small as possible. ...
1
vote
0answers
27 views

Newton's method: Is the change of parameter values between consecutive steps always decreasing?

Assume that I have a twice differentiable function $f(x)$ which I try to maximize with respect to $x$ (let's say $x$ is $k$-dimensional vector). When performing optimization via Newton algorithm, ...
1
vote
0answers
27 views

Lattice fitting to points

I have a set of points (shown as little black circles) which ideally form a hexagonal lattice shape, each point having an equal distance to all of its neighboring points. (Sorry for my drawing, some ...
1
vote
0answers
26 views

Are there any algorithms for simultaneous optimization of multiple objective functions?

I would like to minimize a set of similar objective functions $$f_\boldsymbol{s}(\boldsymbol{x}),$$ where $\boldsymbol{x} \in A \subseteq \mathbb{R}^M$ and the parameterization $\boldsymbol{s} \in S ...
0
votes
1answer
17 views

Why $f^{+}(v)-f^-(v) =val(f)$ if $v$ is the source?

I'm reading Bondy/Murthy's Graph Theory: He defines $x$ as the source and $y$ as the sink, reading a bit later in the chapter, he presents this definitions: $$ f^{+}(v)-f^-(v) = \left\{ ...
0
votes
0answers
29 views

Why does $\int f(x)(y-r(x))\;dP(y,x) = 0$?

My question is, why does: $$\int f(x)(y-r(x))\;dP(y,x) = 0,$$ where $r(x) = \int y \;dP(y|x)$ and $P$ is a probability distribution function. It was also given (in my book) that: $$\int ...
0
votes
0answers
16 views

Thomson problem vs. maximizing sum of distance

Given $N$ equally charged points lying on the unit sphere ("electrons"), the Thomson problem consists of finding the configuration of these points such that the electrostatic potential energy $$ ...
3
votes
1answer
1k views

Linear Programming: Three variable graphical solution

A small bank offers three type of loans: housing loans at $8.50$% interest, education loans at $13.75$% interest rates, and loans to senior citizens at $12.25$% interest. Further, it needs to ...
0
votes
0answers
20 views

Feasible solutions in semidefinite programming - beyond Slater's theorem

Suppose a semidefinite program is given by: \begin{align} \text{Maximize:}&\quad \text{Tr}\left[AX\right],\\ \text{subject to:}&\quad \Phi\left(X\right)=B,\\ &\quad X\geq 0. \end{align} ...
1
vote
1answer
36 views

Formulate the Marriage Problem into a Maximum-flow problem (Graph theory)

Suppose I have $M=\{1,\ldots, n\}$ men and $W = \{1, \ldots, n\}$ women and $B =\{1, \ldots, m\}$ brokers, such that each broker knows a subset of $M \times W$ and for each pair in this subset a ...
0
votes
1answer
45 views

Type of convex function?

I want a convex function $f:\mathbb{R} \to \mathbb{R}$ with the following property: given points $x,d \in \mathbb{R}$, and $\alpha \in (0,1)$, we have $$f(x + \alpha d) \geq \alpha f(x + d).$$ Is ...
0
votes
0answers
17 views

Minimization of the integral with respect to a parameter

Intro Let $f$ be a a real-valued function parametrized by a parameter $\alpha \in \mathbb{R}$ and let $J\colon \mathbb{R} \to \mathbb{R}$ be a functional defined as follows: $$J(\alpha) = ...
0
votes
0answers
21 views

Introduction to Linear Optimization: Driving the artificial variables out of the basis (case: no entries in the $j$-row are nonzero)

Reading the book Introduction to Linear Optimization by Bertsimas and Tsiklisis, I've come across the following subject: Driving the artificial variables out of the basis. The description is as ...
0
votes
1answer
27 views

Is my idea of incoming/outgoing arcs correct?

I'm reading Jungnickel's Graphs, Networks and Algorithms. I've met the following lemma: I know that $e^{-}$ are the incoming vertices and $e^{+}$ are the outgoing vertices. Then I've tried to ...