A non-linear optimization problem includes an objective function (to be minimized or maximized) and some number of equality and/or inequality constraints where the objective or some of the constraints are non-linear. Use this tag for questions related to the theory of solving such problems or for ...

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19
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2answers
508 views

Maximizing curious symmetric function from simple combinatorics

A curious symmetric function crossed my way in some quantum mechanics calculations, and I'm interested its maximum value (for which I do have a conjecture). (This question has been posted at ...
10
votes
1answer
315 views

Bidding Tic Tac Toe

In regular tic tac toe, both the players get alternate chances. This is a variant of that. Player $A$ has $\$x$ amount and player $B$ has $\$y$ amount as initial balance. Assume that $y>x$. Both ...
9
votes
2answers
445 views

A numerical optimization problem with a convolution in the constraint

I have a problem of the following form: minimize $\|Dx\|_2$ subject to $\|x*x\|_2 = 1$ where $x\in\mathbb R^n$, $D$ is a given diagonal matrix of positive entries, and $*$ represents convolution, ...
8
votes
3answers
440 views

Optimization problem for a parity-check code

I have $n$ data blocks and $k$ parity blocks distributed across $m$ boxes where each box can contain atmost $b$ blocks. Each parity block is Ex-or of some data blocks (for ease of understanding we can ...
8
votes
1answer
312 views

Minimization of $\sum \frac{1}{n_k}\ln n_k >1 $ subject to $\sum \frac{1}{n_k}\simeq 1$

Looking at an algorithm for minimizing $\sum_{k=1}^{m} \frac{1}{n_k}\ln n_k > 1$ subject to $\sum_{k=1}^{m}\frac{1}{n_k} = 1$ in which $n_k$ are positive and in general non-sequential integers, I ...
7
votes
7answers
5k 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 ...
7
votes
2answers
252 views

How to solve mixed integer nonlinear programs?

I'm not a math expert so sorry for possible trivial questions. I have written this mixed integer nonlinear program (MINLP): $$ \begin{align} \min & \sum_{i \in ...
7
votes
1answer
229 views

$\max\min$ problem

Suppose there is a real valued function $f(x,y)$ where $x,y$ are vector variables $x\in\mathbb{R}^n$ $y\in\mathbb{R}^m$. Moreover $f$ is strictly/concave in $x$ and strictly/convex in $y$. $f$ is also ...
7
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0answers
123 views

Consider the problem minimize $f(x)= x^4 −1. $

I am studying for a test and I found this problem in the textbook that I'm using, there may be a conceptual problem that I'm coming across. My test is on unconstrained optimization (multi-variable) ...
6
votes
1answer
222 views

Is duality theory in optimization as useful as it seems?

I have been reading a lot about nonlinear optimization and duality, and it seems that duality theory is extremely useful. I feel that I am missing some of the negative aspects/difficulties associated ...
6
votes
3answers
739 views

Finding good approximation for $x^{1/2.4}$

I would like to a good (8 bits accuracy) approximation for $x^{1/2.4}$ in the range $[0, 1]$. This transform is used for converting linear intensities to SRGB compressed values, so it's important that ...
5
votes
1answer
3k views

The composition of two convex functions is convex

Let $f$ be a convex function on a convex domain $\Omega$ and $g$ a convex non-decreasing function on $\mathbb{R}$. prove that the composition of $g(f)$ is convex on $\Omega$. Under what conditions is ...
5
votes
2answers
676 views

Multilinear optimization

Are there any efficient algorithms to solve, multi-linear objective and multi-linear constraint optimization problems? The multilinear functions are sums of bilinear, trilinear (and so on) terms ...
5
votes
3answers
121 views

Find min of $\frac{\left( \sum kx_k \right) \left( \sum x^2_k \right)}{\left( \sum x_k \right)^3}$

With $n \ge 2$ and $x_1,\ x_2,\ \dots,\ x_n > 0$. Find the minimum of: $$ M = \frac{(x_1 + 2 x_2 + ...+ nx_n)( x^2_1 + x^2_2 +...+x^2_n)} {\left( x_1 + x_2 +...+ x_n \right)^3}$$ For specific ...
5
votes
1answer
178 views

Are all non-convex problems created equal?

The distinction between convex and non-convex problems is usually dubbed as the distinction between easy and hard problems. While in the convex case you are golden (local optima are global optima; ...
5
votes
2answers
3k views

Solving an overdetermined system of nonlinear equations

I'm wondering what the "best" way to approach solving a system of the following form would be: $A_1X + Be^{CY} = A_2$ $A_3X + Be^{CY} = A_4$ $A_5X + Be^{CY} = A_6$ etc. EDIT: Coefficients $A_i, ...
5
votes
2answers
1k views

Minimizing with Lagrange multipliers and Newton-Raphson

I am writing a program minimizing a real-valued non-linear function of around 90 real variables subject to around 30 non-linear constraints. I found handy explanation in CERN's Data Analysis ...
5
votes
1answer
749 views

Solving a set of 3 nonlinear equations with constraints

Problem statement: I am given 3 sets of equations that govern the force $P$, and also the neutral axis, defined by two variables, the radius from the center $r$ and also the rotation degree in ...
5
votes
1answer
105 views

Duality theory and nonlinear optimization

I have been studying nonlinear optimization recently and have come across some results that I need clarification for. I will do my best to explain them in detail below, providing citations where ...
5
votes
2answers
78 views

General solution to a system of non linear equations with a specific pattern

I am seeking a general solution to a system of non linear equations with a specific pattern: Order 1: $$ x_0 = a^2 + b^2 $$ $$ x_1 = 2ab $$ Order 2: $$ x_0 = a^2 + b^2 + c^2 $$ $$ x_1 = 2ab + 2bc ...
5
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0answers
213 views

I need help about some compactness arguments

I need help to find some compact sets for my engineering problem. Through this page I learned quite much about it however since I have neither read a book yet nor have an experience I am not able to ...
5
votes
1answer
378 views

How to optimize a rational function

Just a calculus problem: As a function of $K \geq 1$, what is the minimum value of $f/a + f/b + f/c + f/d + f/e$ subject to the following constraints? $$\begin{cases} 1 \leq a \leq c \\ 1 \leq b ...
4
votes
4answers
858 views

Summary of Optimization Methods.

Context: So in a lot of my self-studies, I come across ways to solve problems that involve optimization of some objective function. (I am coming from signal processing background). Anyway, I seem to ...
4
votes
4answers
548 views

Approximate a function over the interval $[0, 1]$ by a polynomial of degree $n$ (or less).

To approximate a function $G$ over the interval $[0,1]$ by a polynomial $P$ of degree $n$ (or less), we minimize the function $f:R^{n+1} \to R$ given by $F(a) = \int_0^1 (G(x) - P_a(x))^2\,dx$, where ...
4
votes
2answers
493 views

Are the Karush-Kuhn-Tucker conditions applicable when one or more of the constraints are nonlinear?

I am just beginning to read about the use of "Concave Programming" methods and use of the Karush-Kuhn-Tucker conditions to identify the maximum value of a non-linear objective function subject to ...
4
votes
2answers
232 views

discontinuous optimization

I'm solving the following problem: $$ \max_\rho \;\; \rho \; \min\left[\left( \frac{bn}{an-bm} \right)[(a-m)-\rho], \frac{b}{a}[a-(p+\rho)]\right]$$ where all constants and variables are defined ...
4
votes
2answers
90 views

Maximize $\prod\limits_{i=1}^n m_i$

Someone visits a market where one hundred different types of fruit are sold. All types cost $1$ euro per pound. The utility the buyer receives from buying $m_1$ pounds of the first type of fruit he ...
4
votes
2answers
185 views

Finding an explicit expression for a minimizer

Suppose $f$ is a continuous function on the interval (0,1). We consider the energy functional $F(u) = \int^1_0\frac{1}{2}((u')^2+u^2)\,dx - \int^1_0fu\,dx$ which is well defined for continuously ...
4
votes
1answer
186 views

Kuhn-Tucker condition is not satisfied

Show that the solution to finding minimum of $f(x)=-x_{1}$ With conditions $-\sin(x_{1})+x_{2} \leq 0$ $x_{1}-x_{2} \leq 0$ is point $(0,0)$, but the Kuhn-Tucker condition is not satisfied in this ...
4
votes
1answer
717 views

Nonlinear Optimization/Programming: A good counter text

I am currently taking a nonlinear optimization course and the text is Bertsekas' "Nonlinear Programming 2e". I think the book does a decent job but I am a much more "hands-on" and visual learner so ...
4
votes
1answer
51 views

How to maximize $\sum\limits_{i=1}^n u_iln(x_i)$?

How to maximize this? $$ \sum\limits_{i=1}^n u_iln(x_i), $$ where $u_i,x_i$ are real numbers, $n$ is a positive integer, $0 \leq u_i \leq 1, 0 < x_i < 1, \sum\limits_{i=1}^n u_i = 1, ...
4
votes
1answer
161 views

Lipschitz continuity of parametric optimizer

Consider the parametric optimal solution $x^{*}: \mathbb{R}^n \rightarrow \mathbb{R}^n$ defined as $$ x^*( y ) := \arg\min_{x \in X } \ \ x^\top x + x^\top A y \\ \quad \qquad \text{subject to: } \ ...
4
votes
1answer
900 views

Linearization of a product of two decision variables

I am trying to solve a problem that involves constraints in which products of two decision variables appear. So far, I read that such products can be reformulated to a difference of two quadratic ...
4
votes
1answer
296 views

Scaling factor and weights in Unscented Transform (UKF)

I'm trying to implement the UKF for parameter estimation as described by Eric A. Wan and Rudolph van der Merwe in Chapter 7 of the Kalman Filtering and Neural Networks book: Free PDF I am confused by ...
4
votes
3answers
102 views

How do one solve a nonlinear combinatoric problem?

I am an undergraduate CS student and I am struggling with a problem. $Qx = b$ where $Q$ is a constant $m \times n$ matrix (with $m>n$), $x$ is a $n \times 1$ vector and $b$ is a $m\times 1$ ...
4
votes
2answers
603 views

Analog of Simplex Method for Quadratic Programming

It happened so I need to work with an algorithm for solving QP problems. The main issue here is that I can't find any references to this algorithms (at least no references in sources in english). ...
4
votes
0answers
97 views

When is the Lagrangian dual function smooth?

Consider a nonlinear optimization problem of the form \begin{align} \min_{x}&\quad f(x)\\ \nonumber \text{subject to } \quad&h_i(x) = 0,\,i=1,\ldots,I\\ \nonumber \quad&g_j(x) \le ...
4
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0answers
38 views

Solving non-linear equations in a chosen subspace

I'm trying to find the root $\mathbf{f(x)=0}$ to the following sets of equations $$ f_1(x,y,z) = x^\prime - \frac{x}{\sqrt{x^2+y^2+z^2}} = 0 \\ f_2(x,y,z) = y^\prime - \frac{y}{\sqrt{x^2+y^2+z^2}} = ...
4
votes
1answer
63 views

QR-Decomposition of matrix valued function

I already posted the following question on MO, but id did not raise much interest there. Maybe the title is too elementary to gain research interest. Suppose I have a matrix valued function $$ ...
4
votes
0answers
33 views

Nonlinear optimisation of Expectation

I am preparing for my exams and I can't get my head around the following question. I know there exists a general method for solving these problems but I don't know where to start. I would greatly ...
4
votes
0answers
176 views

Why Compactness is Necessary at Minimax Theorem

According to Von Neumann's minimax theorem, I have $$\max_{x\in X} \min_{y\in Y}f(x,y)=\min_{y\in Y} \max_{x\in X} f(x,y)$$ for some compact sets $X$ and $Y$ and a convex (in $y$), concave (in $x$) ...
4
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0answers
300 views

(easier version)What is a working example for finding the maximum of a algebraic function $f(a,b,c,d,e)$ with 4 equality …

edit 2: If you are founding an answer would be interesting to you, please upvote this question, because i may use those point in order to start a new bounty. Edit 3: Can somebody answer? Edit 4:I ...
3
votes
2answers
452 views

A naive approach for nonlinear optimization on several real variables and one natural variable. Give examples of (potential) failure

Motivation: Example. To solve a problem on evaluating the maximum of a product of $n$ real variables subject to an equality constraint on its sum $S$ ($=100$), I used the Lagrange multipliers method ...
3
votes
3answers
62 views

Minimization of $log_{a}(bc)+log_{b}(ac)+log_{c}(ab)$?

I am trying to find the minimal value of the expression: $log_{a}(bc)+log_{b}(ac)+log_{c}(ab)$ I think experience gives that the variables should be equal, if so then the minimal value is 6, but ...
3
votes
1answer
2k views

Prove the supremum of the set of affine functions is convex

Let $\langle f_i \rangle _{i \in I}$ be a family of affine functions on a convex and compact set $\Omega \subset \mathbb{R^d}$ such that $f_i = a_i.x +b_i$ for $x \in \Omega$. Prove that f, defined by ...
3
votes
3answers
914 views

Constrained optimization: equality constraint

I have this very general problem (for $n>2$): $$ \begin{align} & \max Z = f(x_1,\ldots ,x_n) \\[10pt] \text{s.t. } & \sum_{i=1}^{n} x_i = B \\[10pt] & x_i \geq 0 \end{align} $$ Assume ...
3
votes
2answers
170 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, ...
3
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2answers
59 views

The quadratic case in nonlinear programming

I'm reading about nonlinear programming and I stumbled into the following statement where I started to wonder a bit: Consider the function $$f(\textbf{x}) = ...
3
votes
1answer
498 views

Optimizing trigonometric and nonlinear functions

First, Please, keep in mind that I'm a programmer not mathematician, and I have a fair mathematical background. I used optimization in Java to fit some observations to a trigonometric function, I ...
3
votes
2answers
179 views

Singular Values of Matrix as Optimization Problem

Assume that $A$ is a positive semidefinite symmetric matrix. It is known that $$\max_{||y||\leq1} \quad y^TAy$$ Has an analytical solution which is the maximum eigenvalue of $A$. This isn't hard ...