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)

2
votes
0answers
31 views

What is this sort of optimisation called?

I am reading a book in mathematical finance. There is something about constrained optimisation. They have specialised it for the financial market, but I am wondering what the general name for this ...
0
votes
0answers
9 views

Strong duality and its relation with perturbation functions

From the strong duality wiki page Strong duality holds if $F^{**}=F$ where $F$ is the perturbation function relating the primal and dual problems and $F^{**}$ is the biconjugate of $F$. I ...
0
votes
0answers
28 views

Split a vector into three

Say we have a vector of length n<100, $v(w_1,w_2,\ldots,w_{n})$. My problem is to divide the vector $v$ into groups of $3$, eg $u_m =(w_i, w_k, w_k)$ with as close weight as possible. Eg to ...
1
vote
1answer
28 views

What does 'the level set is bounded' exactly want to tell?

'The level set is bounded.' occurs in many theorems and other places. I think I can understand the definition of 'level set' but I don't know what does 'it's bounded' want to tell me exactly in ...
0
votes
0answers
9 views

Binary Linear Programm: Check for feasability and multiple solutions

Assuming, I have binary integer program, e.g. given by: $ \arg\min_x \quad 0\\ \text{such that}\quad A_\text{eq} x = b_\text{eq}, x_i \in \{0,1\} $ Where also $[A_\text{eq}]_{ij} \in \{0,1\} $ and ...
0
votes
1answer
35 views

How to show these two problems have equivalent solutions

I have two problems, where $A$ is positive definite: $$\inf \{ x^TAx + b^Tx : 1-x^Tx \ge 0,\ x \in \mathbb R^n\} \ (1)$$ and $$ max_\lambda \ q(\lambda) = -0.25b^T(A+\lambda I)^{-1}b - \lambda : ...
1
vote
2answers
56 views

Derivative of a trace w.r.t matrix within log of matrix sums

I'm trying to solve an optimization (sub)problem and am running into trouble with a tricky derivative. I'd like to find the matrix $C \in \mathbb{R}^{n\times d}_+$ which minimizes the following ...
2
votes
1answer
32 views

How can I show that these two problems have the same optimal solution?

How can I show that these two problems have the same optimal solution: $$\inf \{ x^TAx + b^Tx : 1-x^Tx \ge 0,\ x \in \mathbb R^n\}$$ $$\inf \{ x^TAx + b^Tx : 1-x^Tx = 0,\ x \in \mathbb R^n\}$$ when ...
1
vote
0answers
18 views

Discretization of a convolution integral for constrained optimization problem

I'm working on a constrained optimization problem in which an unknown forcing function, $u(\eta)$, is in the integrand of a convolution integral. To find an optimal shape for $u(\eta)$, the integral ...
-1
votes
0answers
17 views

Optimization problem with embedded absolute values (how to turn to LP)

say I have a problem of the form $$\begin{align*}\min&\sum_i{c_i\,|x_i-|y_i-z_i|+|s_i|\,|}\\[0.3cm] \text{s.t. }&0\le x\le 1\\[0.2cm] &0\le y \le 1\\[0.2cm] &0\le z\le 1\\[0.2cm] ...
0
votes
0answers
22 views

Property of monotone operator (Positive definite)

I would like to prove this statement: "$F$ is monotone if and only if $\nabla F$ is positive semidefinte." I only know $F$ is monotone with respect to $\Omega$ if and only if ...
1
vote
2answers
29 views

Given a fixed volume for a solid cylinder, is it possible to find the minimum or maximum curved surface area?

Given a fixed volume for a solid cylinder, is it possible to find the minimum or maximum curved surface area $ 2 \pi r h $ of this cylinder?
1
vote
0answers
54 views

Prove $\lambda=\min_{i = 1,\ldots, n}\max_{0 \le k \le n-1}\left(\frac {p_i(n)-p_i(k)}{n-k}\right)$

Prove the minimum directed cycle mean cost satisfies: $\lambda = \min_{i = 1,\ldots, n} \max_{0 \le k \le n-1} \left(\frac {p_i(n) - p_i(k)} {n-k}\right)$ using the Bellman-Ford algorithm. Let ...
1
vote
1answer
33 views

Given an optimal solution to the LP, show how it can be used to construct a directed cycle with minimal directed cycle mean cost.

Let $\mathcal G = (\mathcal V, \mathcal A)$ be directed graph with associated edge costs $c_{i,j}$ that has at least one directed cycle. Define the directed cycle mean cost to be $\frac {\{\text {sum ...
1
vote
2answers
34 views

Local optimality of a KKT point.

Consider the problem \begin{equation} \min_x f(x)~~~{\rm s.t.}~~~ g_i(x)\leq 0,~~i=1,\dots,I, \end{equation} where $x$ is the optimization parameter vector, $f(x)$ is the objective function and ...
2
votes
1answer
32 views

How to get the peak value of this logarithmic equation?

Is there a way to get the peak point of the following equation? $$ (a_1-a_2 x)\ln\left(1+\frac{b_1 x}{b_2 x+b_3}\right),$$ where $a_1,a_2,b_1,b_2,b_3$ are all positive constant values and $x$ is also ...
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 ...
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
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 ...
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 ...
1
vote
1answer
74 views

A interesting max min problem

Let $\mathcal{S}\subset\mathbb{R}^2$ be a bounded, closed, compact, convex set which contains origin in its interior. Define \begin{align} c_1^{\star}=\min_{{(x_1,0)\in\mathcal{S}}}~&x_1 ...
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 ...
1
vote
0answers
48 views

Prove an artificial variable that leaves the basis will never return.

Prove an artificial variable that leaves the basis will never return. Edit: This is for the simplex method (I think). I have no idea how to start this. Anyone know any books with these kinds of ...
2
votes
1answer
61 views

Prove optimal solution to dual is not unique if optimal solution to the primal is degenerate

How do I prove an optimal solution to dual is not unique if an optimal solution to the primal is degenerate? I have no idea how to start this. Anyone know any books with these kinds of questions (and ...
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
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 ...
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 ...
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 ...
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 $y$. ...
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!
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
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 ...
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.} \\ ...
1
vote
0answers
33 views

Can we ever have E(argmin(f)) = argmin(E(f))?

Consider a parametric real-valued function $f_{\boldsymbol{\alpha}}:\ \mathbb D^N \rightarrow\mathbb R$ whose parameters $\boldsymbol\alpha$ vary according to some distribution $\psi$, and $\mathbb D$ ...
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: $$f_1(x) = f_1(x_1, x_2,\dots, x_n) = 0,$$ $$f_2(x) = f_2(x_1, x_2,\dots, x_n) = 0,$$ ...
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 ...
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
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. ...