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 ...

learn more… | top users | synonyms

0
votes
0answers
16 views

Epsilon constraint method - Pareto optimal solution representation

There's a course that I do remotely and I have a homework question which I have no idea how to answer. I did look up a lot in google and did not find any good examples - only loads of information and ...
1
vote
1answer
12 views

Non-uniqueness of worst-case (max-min or min-max) optimization

I have a worst-case optimization problem, where i want to maximize the minimum from the uncertainty set (uncertainty is given as an ensemble of 100 realizations, so an ensemble based approach). It is ...
0
votes
0answers
9 views

Can this equation have an explicit solution?

Given $n > 0$, $0 \leq i \leq n$ is an integer, $D = diag(d_1, \dots, d_n)$ is positive definite, $e_i$ is the $i$th column of a $n \times n$ identity matrix, $u \in R^n$ such that $B = D + u * ...
0
votes
1answer
25 views

How do we plot nonlinear differential equations

If this is not nonlinear I apologize, I'm still learning differential equations. I am attempting to make a stream plot of a predator-prey model of eccentric closed curves by using the following ...
-2
votes
2answers
21 views

The optimization problem with max [on hold]

Given $(m\times n)$-matrices $A=(a_{ij})$ and $B=(b_{ij})$, and a vector $c=(c_1, c_2, \ldots, c_m)$; and $\underline{x},\overline{x},\underline{y},\overline{y}$ are real numbers such that ...
-1
votes
0answers
38 views

good extensive library for covex optimization

I need a java library to solve a convex optimization problem, kindly give me few names that are extensively used. I have googled my query but since I am new and have no prior experience in solving ...
0
votes
0answers
34 views

Optimization on fixed sum

Consider this following scenario. Suppose I have $N$ cents, and I want to dispatch these money to $n$ people, each got $x_i$ cents. In order to simplify this problem, we assume the cents are ...
0
votes
0answers
11 views

Dual norm of quasi norms

The dual norm $\Omega^*$ of the norm $\Omega$ is defined for any vector $\mathbf{z} \in \mathrm{R}^N$ by \begin{equation} \Omega^*:= \underset{\mathbf{x} \in \mathrm{R}^N}{max } \quad \mathbf{z}^{T} ...
0
votes
1answer
19 views

How to optimize these parameter

How to optimize the following respect to lambda1 and lambda2: $\sum_{i} f(i) * log(\lambda_1 g(i) + \lambda_2 h(i))$ f(i), g(i), h(i) are known funtions Find lambda1 and lambda2 that satisfy ...
0
votes
1answer
13 views

Necessary Conditions for Saddle Value point

This questions is from the Kuhn-Tucker paper "Nonlinear Programming" in Section 2 Lemma 1. I don't understand how those conditions are necessary for a saddle point. I always thought that a saddle ...
0
votes
0answers
19 views

Binary Integer Programming with one nonlinear constraint [on hold]

I want to solve the following problem : That xi,j∈{0,1} and yi∈{0,1} are binary optimization variables and ai ,bi have defined value. This optimization is Binary Integer Linear Programming and can ...
-3
votes
0answers
35 views

How to convex problem with multiple constraints? $\underset{\mathbf{A}}{\text{minimize }} \|\mathbf{A}\|_F^2$

IF $\mathbf{X=AS}$ where $\mathbf{X} \in R_+^{n \times m}$, $\mathbf{S} \in R_+^{r \times m}$ are known variable and $\mathbf{A} \in R_+^{n \times r}$ is unknown variable. here $R_+$ denotes the set ...
0
votes
0answers
36 views

Non-convex function with global minimum [duplicate]

I am working on a complicated objective function which I suppose is not convex. But when I use a global optimization tool that can find all its local minimums, it will always converge to the same ...
0
votes
1answer
36 views

Maximum likeliood estimation of variances of transformed variables

I use MATLAB's fminunc function in order to find the minimum of a negative log-likelihood function $f(\overrightarrow{\theta})$, parametrized by 3 parameters lets say ...
0
votes
0answers
24 views

How can I minimize a quadratic on the unit simplex?

How can I compute $$ \min_{x \in \Delta_n} \frac{1}{2}\lVert Bx\rVert^2 + x^tAy$$ with $x \in \mathbb{R}^n, y \in \mathbb{R}^m, A_{m \times n}$, $B_{n \times n}$ where $\Delta_n$ is the unit simplex ...
1
vote
0answers
15 views

Maximizing the frobenius norm subject to constraints $\underset{\mathbf{S}}{\text{maximize }} \|\mathbf{S}\|_F^2$

IF $\mathbf{X=AS}$ where $\mathbf{X} \in R_+^{n \times m}$, $\mathbf{A} \in R_+^{n \times r}$ are known variable and $\mathbf{S} \in R_+^{r \times m}$ is unknown variable, How to solve the below ...
0
votes
0answers
18 views

KKT Sufficient condition when optimal solution is intuitively at the boundary

My optimization problem is: $\operatorname{arg\,max}_P \sqrt P$ subject to $P \le \upsilon_\tau$ where $P \in \mathbb{R}^+$ and $\upsilon_\tau \in \mathbb{R}^+$ Intuitively, because $\sqrt P$ is ...
0
votes
0answers
8 views

optimization of formulas involving binomial coefficients

I encountered such a problem. We need to find the min value and max value of $f(x,y)$. $x$ and $y$ are integers $\in[0,n]\times[0,n]$ and $(x,y)\neq (0,0)$ or $(n,n)$. $$ ...
0
votes
0answers
18 views

How can I solve $\min \{ \langle A(x),y\rangle + f(y) \text{ s.t. } y \in S^n, \operatorname{tr}(y) =1, y \geq 0\}$?

I'm trying to solve the problem $$\min \{ \langle A(x),y\rangle + f(y) \mid y \in S^m, \operatorname{tr}(y) =1, y \geq 0\}$$ where $x \in \mathbb{R}^n$, $y \in S^M$, that is, it's a symmetric $m$ by ...
1
vote
0answers
158 views

What is the nature and location of maxima of expected log-utility?

Is is possible to describe, and locate efficiently, the maxima of the function below in the parameters $\mathbf{x}$ $$\sum_{i} p_i \log( N + \sum_j x_j[B_j +(A_j-B_j)\delta_{ij} + min(A_j,B_j) ]) ...
0
votes
2answers
42 views

Minimize a non-convex function subject to linear dynamics constraint

I want to solve the following problem: $$\min\limits_{\bf u} \frac{\bf c^T {\bf x} (T_f)}{\| \bf c\|\|{\bf x} (T_f)\|}$$ subject to $$\dot{\bf x} (t) = A {\bf x}(t) + B {\bf u}(t)$$ $$x(0) = x_0$$ ...
0
votes
1answer
29 views

Find a line such that sum of perpendicular distances of points to the line is minimized

Given a set of points (column vectors) $S = \{p_1, p_2, \cdots, p_n\} \subset \Re^d$, let $A \in \Re^{n \times d}$ be a matrix of which each row is just $p_i^T$. It is easy to find a unit vector $s_1$ ...
1
vote
2answers
45 views

Is the optimum of this problem unique?

I have a convex optimisation problem: $$\min_{x_i} \sum_{i=1}^N a_i \exp(-x_i/b_i)\quad\text{ s.t. }\sum_{i=1}^N x_i=x\text{ and } x_i\ge 0$$ The KKT conditions are: $$\lambda=\begin{cases} ...
0
votes
1answer
53 views

Solving non-linear optimization using generalized reduced gradient (GRG) method

Consider the following elementary maximization problem: \begin{align} f{=}\mathrm{argmax}_{y_{l,c}, p_{l,c}}~\sum_{l=1}^{L}\sum_{c=1}^{C} y_{l,c}\text{log}_2\left(1+\frac{p_{l,c}}{I_{l,c}}\right) ...
0
votes
0answers
34 views

Question about duality in nonlinear optimization

Let $f(x)$ and $h(x)$ be functions from $\mathbb{R}^n$ to $\mathbb{R}$ and consider the minimization problem $$ {\rm minimize} ~~~ f(x)$$ $$~~~~~~~~~{\rm subject ~to}~~h(x)=0.$$ Suppose the minimum is ...
2
votes
1answer
46 views

Minimize Product of Sums of Squared Distances (Also a tensor problem)

The Question Given two sets of vectors $S_1$ and $S_2$,we want to find a unit vector $s$ such that $$\{\sum_{u\in S_1}(\|u\|^2-\langle u, s \rangle^2)\} \cdot \{\sum_{v\in S_2}(\|v\|^2 - \langle v, ...
0
votes
1answer
21 views

Solving three-variable nonlinear equation systems

A physical problem which I've been studying leads to the following nonlinear equation system to be solved: $$\alpha\cdot79\cdot A_1 +(1-\alpha)\cdot 1025 \cdot B_1 = C_{11}$$ $$\alpha\cdot145\cdot A_1 ...
0
votes
2answers
25 views

How to find more than two coefficient for single variable nonlinear equation?

I don't have good knowledge on mathematics, but now I faced one problem with maths. That is, I have a data set which contains only one independent and one dependent variable. Now I have a equation ...
1
vote
1answer
33 views

how to write largest circle inscribed inside a triangle as an optimization problem?

can someone show me how to write this problem as a convex optimization problem.Find the largest disk that can be bounded by $X \geq 0$ , $Y \geq0$ and $X+2Y\leq1$. My institution is to cast to ...
1
vote
1answer
40 views

Minimize the squared dot product of two specific vectors

Do you think there exists a efficient algorithm(non brute-force) for the following problem. I search the optimal solution for the following problem: Given a vector $u=(u_1, u_2,..., u_k)^T$ with ...
1
vote
1answer
49 views

Is standard eigenvalue optimization problem convex

For any arbitrary symmetric matrix A , is the standard eigenvalue problem convex $ \lambda_{max}(A)= \max_{\|x\| \leq1} x^{T}Ax$
1
vote
0answers
31 views

Tucker's theorem from Farkas lemma

I am trying to understand the proof of Tucker's theorem using Farkas lemma but there are some points that are not clear to me. The proof I am following is in this paper at page 16. What I do not ...
3
votes
1answer
45 views

Non-linear system vs minimisation problem

If you have a non-linear system of equations which can be formally written as : \begin{equation} \begin{cases} F_1(\mathbf{x})=0\\ F_2(\mathbf{x})=0\\ \ \ \ \ \vdots\\ F_n(\mathbf{x})=0\\ \end{cases} ...
0
votes
0answers
11 views

Broyden's Method mismatched dimensions

Compute the first two iterates $\mathbf{x}^{(1)}$, $\mathbf{x}^{(2)}$ using Broyden's Method for the initial point $\mathbf{x}^{(0)}=(1,4)$ and the function $f(x_{1},x_{2})=3x_{1}^2+x_{2}^2-x_1x_2$ ...
1
vote
2answers
47 views

Performing Second Derivative test on multivariate function

I have two functions $f=xy^2$ and $g=x^2+y^2$. When optimizing $xy^2$ on the circle $x^2+y^2=1$ I get 6 critical points but when I try to perform the second derivative test, it equals 0, meaning that ...
1
vote
0answers
26 views

what is the good book to learn nonlinear programming

I would like to learn nonlinear programing. what is the best book to do so and I prefer if the solution manual of the book is available. thanks
1
vote
1answer
101 views

Optimization over vector spaces. Generalized KKT.

I am looking for the extension of the theorem I found in the book by Luenberger called "Optimization by vector space methods." Here is the statement of that theorem from Luenberger: Generalized ...
0
votes
1answer
22 views

Calculating the Hessian for nonlinear objective functions

In least squares minimization, we minimize the sum of squares of the residuals: $f = ||b-Ax||_2 ^2 $ The Hessian of this matrix is given by $2A^TA$, and from the Hessian we can do the second partial ...
1
vote
0answers
33 views

How to solve coupled nonlinear ODEs with a algebraic constraints?

Here is the problem I'm currently facing right now, I have set of ODEs which I can solve numerically given the initial condition. But I'm not sure how to go about giving algebraic solution with ...
0
votes
0answers
11 views

Conditions of expression to be positive

Let $f:t\mapsto f(t)$ defined as follow: $$ f(t)= - p\, c + \frac{a\, p }{1 + g(t)^n} - \frac{a\, n \, g(t)^n}{(1+ g(t)^n)^2}; $$ Condition 1: $ a > 0, \, c > 0\, , a > 2 \, c, \, ...
1
vote
1answer
23 views

Prove a specific Cartan matrix is positive definite

I am trying to prove that the following matrix is positive definite, but I am stuck in the last step of my proof... Any help would be really appreciated. Thanks! Question Let $A$ be a matrix with ...
0
votes
1answer
12 views

Can a symmetric, positive-definite, real matrix with only 1s on the main diagonal have an off diagonal element with absolute value greater than 1?

I am working with correlation matrices and I would like to know if every symmetric, positive definite matrix with 1s on the main diagonal is a correlation matrix (i.e. all its off diagonal elements ...
0
votes
0answers
56 views

Optimizing concave function over non-convex set

I have the following problem that I am looking advice on. Let $ \mathcal{F}$ be a convex subset of vector space $X$. The goal it to \begin{align*} \max_{x \in \mathcal{F}} f(x)\\ s.t. \ g(x) \le 0 ...
0
votes
2answers
54 views

Explaining the “well-known” optimization of this particularly simple convex, non-differentiable function?

I've been programming algorithms for solving L1-regularized logistic regression with large datasets. As such, I've been delving into the computer science literature, and came across the following ...
0
votes
2answers
58 views

What is the merit function?

When we use merit function in optimization & why uses this function? if we use merit function the space must be convex or not?
0
votes
1answer
53 views

How to solve this problem efficiently?

I have this problem \begin{align} \min_{\alpha,\beta,X}~&<\alpha \cdot X+\beta \cdot Y,D>-c \cdot (<\alpha \cdot X+\beta \cdot Y,H>)^{1/2}\\ &X,\alpha,\beta>=0\ ...
0
votes
1answer
29 views

what is the trust region algorithm in optimization?

I see some books that say the trust region work with contour's line .but i can't understand how choose the point with contour's line and sort them? thank you if answer me.
1
vote
0answers
15 views

Basis pursuit denoising

The Lagrangian form of basis pursuit denoising $\min_{w} ||w||_{\ell_{1}} + \lambda ||Aw-x||_{\ell_{2}}^{2}$ can be solved using proximal gradient descent. Proximal methods also can be used to ...
2
votes
1answer
19 views

Showing that $x_{k+2}$ is a point which approximates a maximum?

Suppose that $x_k<x_{k+1}$ and $f'(x_k)>f'(x_{k+1})$. How can I show that the secant method will give $x_{k+2}$ as a point which approximates a maximum? $$x_{k+2}= ...
1
vote
0answers
24 views

Minimum of the difference of two logarithms

I am trying to find an analytical expression of the minimum of $$ f_n(x) = \frac{2x}{n^2+n}\log(x) - \frac{2x+2}{n^2+3n+2}\log(x+1) $$ when $x\in [1;n]$ I used to think from graphing it that this ...