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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3
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1answer
129 views

minimizing $\sum_{i=1}^n \max(|x_i - x|, |y_i - y|)$

Suppose there are $n$ points $(x_i, y_i)$ for $i = 1,\ldots,n$. Please find another point $(x, y)$ to minimize function: $$\sum_{i=1}^n \max(|x_i - x|, |y_i - y|)$$
2
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1answer
185 views

Control on Conformal map

Let $\Omega$ be smooth simply connected open set of $\mathbb{R}^2$ such that $\overline{\Omega}$ is compact. We know that there exists a conformal diffeomorphism $\psi$ from $\mathbb{D}$ to $\Omega$. ...
7
votes
1answer
4k 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 $...
3
votes
1answer
3k 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 ...
0
votes
2answers
112 views

Constraint minimization of sum of Non-symmetric matrices

I am trying to find closed form solution to following problem \begin{equation} \begin{array}{c} \text{min} \hspace{4mm} \big(\lambda_1\left( \mathbf{y}^T V^{(1)}\mathbf{x} \right)^2 + \lambda_2\...
8
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7answers
7k 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 ...
8
votes
3answers
472 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 ...
6
votes
2answers
339 views

Optimal path around an invisible wall [duplicate]

The Problem On an infinite plane there are two points, $A$ and $B$, a unit distance apart. There is a $50\%$ probability that there is an invisible wall somewhere between the two points. The ...
5
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2answers
942 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 \...
2
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1answer
1k views

When $\min \max = \max \min$?

Let $X \subset \mathbb{R}^n$ and $Y \subset \mathbb{R}^m$ be compact sets. Consider a continuous function $f : X \times Y \rightarrow \mathbb{R}$. Say under which condition we have $$ \min_{x \in X}...
8
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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 ...
5
votes
2answers
99 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 $...
4
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1answer
2k views

Intuition for gradient descent with Nesterov momentum

A clear article on Nesterov’s Accelerated Gradient Descent (S. Bubeck, April 2013) says The intuition behind the algorithm is quite difficult to grasp, and unfortunately the analysis will not be ...
3
votes
0answers
111 views

If this problem is not unbounded, what's wrong in this dual derivation?

In a paper with 100 citation, Robust Support Vector Machine Training via Convex outlier Ablation, a convex relaxation is used. In this paper, a form of robust svm proposed: \begin{align} \min_{0\leq \...
3
votes
1answer
313 views

Newton optimization algorithm with non-positive definite Hessian

In the newton optimization algorithm to find the local minimum $x^*$ of a non-linear function $f(x)$ with iteration sequence of $x_0 \rightarrow x_1 \rightarrow x_2 ... \rightarrow x^*$ all $\nabla ^2 ...
2
votes
1answer
382 views

Does the uniqueness of solutions to convex optimization with linear constraints hold in n>3 dimensions?

This is a repost of an earlier question, where I think I was not clear enough in what I was asking: I am examining the following optimization problem, for which I would like to know if, when a ...
2
votes
1answer
2k views

Optimization of a function of two variables.

Suppose we have to minimize a function $f(\mathbf x,\mathbf y)$ where $\mathbf x$, $\mathbf y$ are vectors in Euclidean space. The function is convex in $\mathbf y$ when $\mathbf x$ is kept constant ...
1
vote
1answer
83 views

Reformulation of BQP to SDP

I run into the following reading some optimization papars: $$\min_x x^TAx $$ where $x\in\{-1,1\}^n$ and $A\in S_n$, Is equivalent to $$ \min <X,A>$$ s.t $diag(X) = (1,1,...,1)\;\; rank(X) = 1$. ...
1
vote
1answer
208 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
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3answers
461 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 $g_i(...
0
votes
1answer
25 views

Optimization involving integrals with varying limits

What are the common methods and tools to tackle optimization problemsinvolving integrals. To be precise lets consider the following optimization problem that I came across with: $$\text{maximize}\,\,F(...
4
votes
0answers
320 views

Optimizing non linear programs of two variables

The scenario is; We've got $n$ stationary 360$^{\circ}$ sensors in an confined area (each sensor is located at some arbitrary $\left(x,y\right) = \left(x_{n},y_{n}\right)$), once a unit $t$ enters ...
4
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0answers
202 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 $x$) ...
3
votes
1answer
316 views

Finding a function that satisfies constraints numerically

I have the following system of equations for function $p(y)$ and I need help debugging my solution: $$\begin{align} 0&=\log(p(y))+1-\lambda-\gamma y^2-\eta \left(\frac{e^{-y^2/2}}{\sqrt{2\pi}}\...
3
votes
1answer
213 views

non linear optimization

How to solve this optimization problem using matlab or some other tool. I know that, this is a convex problem with non-linear constraint $\rho\geq \rho_{min}$ , so i have tried many times it in ...
2
votes
2answers
941 views

Classifying local behavior of fixed points using eigenvalues from linear stability analysis of 3D system

I've learned about classification of fixed points of 2D systems using linear stability analysis and I am wondering how if at all I can apply the same process to analyzing local behavior of fixed ...
2
votes
1answer
600 views

Difference: Newton's method, Newton-Rhapson method, Gauss Newton-method.

I would appreciate some clarification w.r.t. algorithms for solving nonlinear systems of equations. 1 - I don't understand the difference between Newton's method and Newton-Rhapson method. In [1], ...
2
votes
2answers
726 views

How to maximize an entropy function?

I'm very novice in optimization and have a convex optimization function of form $\sum_{i,k} p_{k,i}*\log{p_{k,i}} $ to minimize with the following constraints: $\forall i, a_i = \sum_{k=1}^{m} b_k. ...
1
vote
2answers
354 views

Why nonlinear programming problem (NLO) called “nonlinear”? What does “nonlinearity” actually mean? Is it “not linear” or something different?

My teacher in the course Mat-2.3139 presented the same definition as in Wikipedia for the nonlinear programming problem here but he did not specify what the nonlinearity actually means or what it ...
1
vote
2answers
164 views

Find the range of $x$, given $y_{min} \leq y(x) \leq y_{max}$, where $y(x) $ can be any function ( Updated)

I have a series of inequalities: $$y_{1min} \leq y_{1}(x) \leq y_{1max}$$ $$y_{2min} \leq y_{2}(x) \leq y_{2max}$$ $$..$$ $$y_{nmin} \leq y_{n}(x) \leq y_{nmax}$$ Note that $x\in\mathbb{R}$ The ...
1
vote
1answer
509 views

Properties of the positive definite Hessian matrix of a convex function

I'm reading about nonlinear programming and I'm having trouble understanding the cool properties that a positive definite Hessian matrix $Q$ of $n$-dimensional function $f: \mathbf{R}^n\rightarrow \...
1
vote
5answers
489 views

Independence of Rotation Matrix Definitions

I am trying to solve a system of non-linear equations. I know that 9 of my variables put together form a 3x3 rotation matrix $$ A = \left( \begin{matrix} a_{11}& a_{12}& a_{13}\\ a_{21}& ...
1
vote
1answer
59 views

A non-linear optimization problem

I have the following optimization problem on the variables $a_1, ..., a_n$: $$ minimize \frac{\sum_{k=1}^{n}\max(k\cdot a_{k},1)}{\sum_{k=1}^{n}a_{k}} $$ $$ such\ that\ \ 0\leq a_k\leq 1\ \ \ (k=1, .....
1
vote
2answers
280 views

What is a convex optimisation problem? Objective function convex, domain convex or codomain convex?

My teacher in the course Mat-2.3139 did not want to answer this question because it would take too much time. So what does a convex optimisation problem actually mean? Convex objective function? ...
1
vote
1answer
70 views

A minimization problem

Define $$L(w,u)=\frac{1}{2}\|w-u\|^2+\beta \left\|\frac{w}{x}\right\|,~w,u\in \Bbb{R}^n$$ where $$\frac{w}{x}=\left(\frac{w_1}{x_1},\ldots, \frac{w_n}{x_n}\right)$$ $$\|x\|=\sqrt{x_1^2+\cdots+x_n^2}$$ ...
1
vote
1answer
89 views

Constrained optimization problem of 4 variables!

I am stuck with this problem. I thought of trying to first solve the problem with weak inequalities for all the constraints using Kuhn Tucker conditions, and checking for solutions at which the ...
0
votes
0answers
26 views

Are the constrained optimization problem equal to the unconstrained one?

(1) \begin{equation}\label{constrained} \begin{array}{cl} \arg \min \limits_{x} & \|Ax-b\|_2\\ \mathrm{s.t.} & \|x\|_1\le \epsilon \end{array} \end{equation} (2) \begin{equation}\label{...
0
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0answers
178 views

MiniMax Theorem

Consider the compact sets $X \in \mathbb{R}^n$, $Y \in \mathbb{R}^m$, $A \in \mathbb{R}^n$, $M \in \mathbb{R}^{n \times m}$. For fixed $(\bar{a},\bar{B}) \in A \times M$, by the MiniMax Theorem we ...
0
votes
1answer
158 views

An Interesting Resource Allocation Problem

Here is the problem: \begin{array}{ll} \text{minimize} & \sum_{i=1}^N \frac{1}{1 + \textrm{exp}(C_i + x_i)}\\ \text{subject to} & \sum_{i=1}^N x_i \le R \\ & x_i \ge 0, ~ i = 1,2,...,N \...