2
votes
1answer
39 views

Origin of Slater's condition

I've been looking all over the internet to answer this question: Slater's condition is a commonly used to certify that strong duality holds in a convex optimization problem. Although used in many ...
1
vote
1answer
23 views

Linear Optimization Study Material

I've recently enrolled in a linear optimization course, and it's been a while since I've taken linear algebra. I do not yet have access to the book for the course or I would skim it to see what I need ...
2
votes
1answer
54 views

Got stuck with this $L^2(-1, 1)$ optimization problem. Any ideas where it comes from?

Statement Let $u(x) \in L^2(-1, 1)$. Solve the following optimization problem: $$ \begin{cases} J(u) = 4 \int_{-1}^{1} \sqrt{|x| (1 - |x|)} u(x) dx + \left(\int_{-1}^1 \sin(3\pi x) u(x) dx\right)^2 ...
1
vote
1answer
15 views

Placing a shape on a grid

I am interested in a certain kind of geometrical optimisation problems. I will illustrate it on a semi-concrete example: You are given a two-dimensional shape, say a polygon, and a rectangular ...
0
votes
1answer
22 views

Literature study for Optimal Estimation Theory

It seems Optimal Estimation/Control Theory requires a lot more than undergraduate maths. Any good book that would help me get started? I have so far referred the following books but found them quite ...
0
votes
1answer
71 views

Optimization Problem Reference Request

I am interested in learning if the following is a canonical problem, or if there is a good reference for it. It seems like something that might appear in a textbook, but you never really know with ...
0
votes
1answer
45 views

Constrained optimization with complex variables

Is there a theory of constrained optimization with complex variables, do you know any textbook on that topic? The typical textbooks on constrained optimization deal with real variables. I actually ...
2
votes
0answers
65 views

Algorithm to find maximum possible value of the minimum expression in a list

Problem Let $x_1, x_2, \ldots, x_m \in \mathbb{R}$, and suppose I have a bunch of expressions which are linear combinations of $1, x_1, x_2, \ldots, x_m$. For example, I might have \begin{align*} ...
6
votes
0answers
531 views

Optimization / personalization within clusters

I have the following optimization problem: I have a (random and very noisy) objective function f(A, P), where A is a vector of "observable" parameters of the input and P is the parameters that I can ...
3
votes
0answers
54 views

Operational Research. (Ressource Management)

I am looking for a solution that i know exists already in the field of "Operational Research"... I Just can't put my finger on the name of the thing. An heuristic to solve a very common and simple ...
1
vote
0answers
17 views

Internals of a MIP Solver

I would like to learn about the internals of a Mixed Integer Programming (MIP) solver. Which concepts shall I read about? Are there a couple of standard books which can be a good start?
1
vote
2answers
84 views

What is this mathematics sub-field called?

I would love to answer another question on this site, but I am totally unfamiliar with the required technique. I mean, I don't even know the sub-field's name. The field I am looking for is one that ...
1
vote
1answer
88 views

What's a really good book for a course titled “Optimization and Control Theory”?

I can't seem to find one that shows a lot of examples with the theory. Could I get some help? Also, it would be a bonus if the book/material is readily available online so I can download it onto my ...
0
votes
1answer
110 views

Reference request - second derivative test for function of two variables that includes details of what you can infer when discriminant is zero

The second derivative test for functions of two variables as I have learned and taught in calculus classes says, in part, that if at a point $D=f_{xx}f_{yy}-(f_{xy})^2$ is zero then we can tell ...
0
votes
1answer
41 views

Approaches to fitting noisy oscillatory data?

I have observations $\hat{f}$ from data at points $\mathbf{x}=\{x_1,\ldots,x_N\}$, that is modeled as a known oscillatory form $f(k\ x)$ (for example, the sinc function), where $k$ controls the ...
1
vote
1answer
75 views

Multi-objective optimisation methods

I have recently encountered a few problems concerning the optimisation (maximization or minimization) of one or more functions under some constraints. Are there any good introductory tutorials or ...
1
vote
0answers
45 views

Conditions for the ground state of Gibbs ensemble not to be “degenerate”

I am looking at the Wikipedia article on Partition function -- As a measure. Unfortunately the article has no relevant references or reading suggestions. I am looking for books or other resources ...
1
vote
0answers
54 views

Suggestions for a reference-level text on optimization theory?

I'd be interested in knowing if anybody has suggestions on an advanced but still self-contained reference on optimization theory, centered around linear and convex problems. The key feature of my ...
3
votes
1answer
94 views

Simple optimization trick

Let $f,g:X\to\Bbb R$ be two functions where $X$ is any set. Then $$ \left|\sup_x f(x) - \sup_x g(x)\right|\leq \sup_x|f(x) - g(x)|. $$ This fact is fairly easy to prove, but it seems to be a ...
1
vote
3answers
390 views

A constrained linear least Frobenius norm problem:$\min_{X} \|A-XB\|_F$ subject to $Xv=0$?

Assume we are given two matrices $A, B \in \mathbb R^{n \times m}$ and a vector $v \in \mathbb R^n$. $\|\cdot\|_F$ is the Frobenius norm of a matrix. How can we solve $$\min_{X \in \mathbb R^{n ...
4
votes
1answer
118 views

References on constrained least square problems?

I have met some constrained least square problems, for example, my last post. I found that there are various methods for slightly different constraints, and still I often had little clue about how to ...
1
vote
1answer
70 views

Reference request, discrete optimization problem

i am not an expert in optimization. I have stumbled across an interesting problem which looks to me like there should already exist literature about it. The problem can be phrased as follows: given ...
1
vote
1answer
110 views

“Cookbook” methods for neighborhood structure design in simulated annealing for combinatorial optimization?

What are some "cookbook" methods for neighborhood structure design in simulated annealing for combinatorial optimization? Are some reviews or books that contain some "cookbook" methods for ...
1
vote
0answers
718 views

Simple example application of Karush-Kuhn-Tucker conditions to minimization problem

I am wondering if there is a simple example application of the Karush-Kuhn-Tucker conditions to show that a minimum exists for a multivariate minimization/optimization problem. Could anyone suggest a ...
2
votes
1answer
133 views

Book on constrained numerical optimization

For unconstrained numerical optimization I have been using the book "Numerical Methods for Unconstrained Optimization and Nonlinear Equations" by Dennis and Schnabel. I found it to be a great book ...
0
votes
1answer
46 views

Degeneracy of the analytic center of a set of linear inequalities

I have a question about the degeneracy of the analytic center of a set of linear inequalities. When the set of linear inequalities is degenerate, I guess that the analytic center would also be ...
3
votes
1answer
56 views

Convex programming when the problem has an underlying combinatorial structure that's a DAG

I have a nonlinear convex objective function to minimize. The function is defined on a set of variables: $\{ x_1,x_2, \ldots ,x_p \},$ where each $x_i$ is a number associated with a path in the DAG. ...
1
vote
0answers
234 views

A convex programming problem involving sum of logarithms of linear functions

Here is a convex programming problem I encountered while working on an estimation problem for a mixture of multinomial distributions. We have a matrix $A_{m \times n}$ containing non-negative real ...
1
vote
0answers
41 views

What's is this optimization function property called?

An optimization function $$f:\bigcup_{n \in \mathbf{N}} S^n \to \mathbf{R}$$ may have the property that given a domain $$P = \prod_{i=1}^N P_i$$ and the solution $$(m_i)_{i=1}^N = \underset{x \in ...
1
vote
3answers
547 views

Introductory optimization book

Does anyone knows a book about optimization that starts from the very basic calculus optimization, i've searched for it but they sometimes assume you have that basic knowledge, starting from linear ...
3
votes
2answers
154 views

Maximum of a product of a polynomial with positive coefficients and a finite sum of exponentials with negative coefficients on $[0,+\infty)$

Prove or disprove that $$ f(x)=\left(\sum_i a_i x^i\right)\left(\sum_j b_j e^{-\lambda_j x}\right) $$ where $\forall i, a_i>0$, $\forall j, b_j>0,\lambda_j>0$, and both sums are finite, ...
1
vote
0answers
79 views

Interval Algorithm for Gradient Descent Method

Are there any references discussing an interval algorithm for the vanilla gradient descent method given a function $f \colon \mathbb{R}^n \to \mathbb{R}$? Edit: In particular, I am searching for an ...
2
votes
2answers
147 views

Dynamic Optimization - Infinite dimensional spaces - Reference request

Respected community members, I am currently reading the book "recursive macroeconomic theory" by Sargent and Ljungqvist. While reading this book I have realized that I do not always fully understand ...
1
vote
1answer
52 views

How to construct a polynomial with minimum deviation from zero on the complex region?

I need to compute the analog of Chebyshev polynomials (which give the minimum deviation from zero on [-1,1]) on the given region $\Omega\subset \mathbb C$. More precisely: find $P_n$ such that ...
4
votes
1answer
179 views

Book on advanced topics of Network Flows

I am taking linear optimization class. Could you suggest me good fundamental textbook on advanced topics of network flows. To be more specific I am interested in: Multicommodity flow and multicut, the ...
1
vote
1answer
160 views

Voronoi diagram with different metric functions

Given a metric space $(X,d)$ and finite number of points $(x_i)_{i=1}^n$ the Voronoi diagram (or the Dirichlet cell) $C_i$ is given by $$ C_i = \{x\in X:d(x,x_i)<\min\limits_{j\neq i}d(x,x_j)\}. ...
2
votes
1answer
59 views

Specific solvable cases of TSP

Did a quick search on polynomial time solvable TSP and found some references such as this one for special cases for the bottleneck TSP. Was wondering if anyone was aware of any references that catalog ...
10
votes
4answers
2k views

Looking to understand the rationale for money denomination

Money is typically denominated in a way that allows for a greedy algorithm when computing a given amount $s$ as a sum of denominations $d_i$ of coins or bills: $$ s = \sum_{i=1}^k n_i ...
7
votes
6answers
3k views

Linear Programming Books

Do you know of a good book on linear programming? To be more specific, i am taking linear optimization class and my textbook sucks. Teacher is not too involved in this class so can't get too much help ...
2
votes
0answers
258 views

Sharp (Reverse) Harmonic-Arithmetic Mean Bounds

Let $\mathbf{x} =$ {$x_{i}$} be a set of $n$ positive reals. In every good book on inequalities, one finds the classical result \begin{eqnarray} AM(\mathbf{x}) \geq GM(\mathbf{x}) \geq ...
6
votes
7answers
2k views

Operations research book to start with

for somebody having a quite strong background in Mathematics, which are some good books for the domain of Operations research? I guess there are textbooks covering topics like linear and nonlinear ...