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

1
vote
1answer
26 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 ...
0
votes
0answers
18 views

Handle (boolean/integer) decision-variables in heuristic optimization algorithms [on hold]

Decision variables (at least in my problem) tend to change the problem in a rather drastic way. The different states control like on-and-off parameters how the objective function/or constraints are ...
1
vote
1answer
35 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
44 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
24 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
43 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
44 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
22 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
92 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
18 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
27 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
votes
0answers
18 views

Regression for matrix [on hold]

I have a non linear MIMO state system (dx/dt=Ax+Bu;y=Cx+Du) whose quite a large number of input output pairs are known, but its exact transformation formula (A,B,C,D) is unknown. I want to do a linear ...
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 ...
-1
votes
0answers
14 views

linearizing a non linear optimization problem

I have a mathematical model with an objective function which is non-linear and some linear constraint. In objective function I have the term xy, that x and y are continuous and both x and y are ...
0
votes
1answer
9 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
53 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
50 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
51 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
28 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
14 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}= ...
-2
votes
0answers
45 views

how can I find the minimum of $f(x)$ by bisection method? [on hold]

what will happen if the bisection method is applied in the range $a\le x\le b$ when $f(x)$ does not have a minimum in this range?
-1
votes
0answers
15 views

In Fermat-Weber's problem, show that if $x$ is not an anchor point, then $\nabla f(x)$ will exist

I have this problem which I think I have kind of solved, but I don't know how to calculate the derivative of the following $f(x)$ without using the formula $\nabla ||u||=\frac{u}{||u||}$ and I also ...
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 ...
0
votes
0answers
33 views

Strict Feasibility in Interior Point Methods [on hold]

As we know, in the interior point methods, all the iterates have to be strictly feasible. I implemented an affine scaling interior point for nonlinear objective functions. For small examples (2D), it ...
0
votes
0answers
28 views

Indicator Variable, Mixed Integer Linear Programming

Assume $x$ is a real variable, and $0\leq x \leq1$. Besides, $y$ is a binary random variable. I need a linear program that: if $y$ is $1$: $x>0$, if $y$ is $0$: $x=0$ I know the following ...
0
votes
1answer
30 views

is this a convex optimization problem?

Can someone clarify is this a convex optimization problem or not. $min \| X-UV\|_{F}\quad $ s.t $ \quad U \geq ,V\geq0$ . If not , what makes the problem non-convex?
0
votes
1answer
55 views

How to perform the optimization when gradient is a matrix $\mathbf{R}^{n\times n}$

I am trying to optimize this cost function by using Gauss-Newton method. $$f = \sum_{i = 1}^n Tr{(Z^TZ)}$$ where $Z$ is a $4\times4$ matrix and it is a function of real vector ...
0
votes
0answers
16 views

relationship of eigenvalues of a matrix with conjugate gradient method

Assume that $Q$ has all its eigenvalues in the two intervals $[a,b]$, $[a+\delta,b+\delta]$, while $a,b,\delta>0$. Show that for every start point $x_0$, after two steps of conjugate gradient ...
0
votes
0answers
49 views

Nonsmooth optimization

Now I have a chance taking a course in nonsmooth optimization, the course outline writes: convex analysis, subdifferential calculus and proximal mapping. various numerical algorithms to solve ...
1
vote
1answer
21 views

Proving a function of matrix is convex

I have a function of a matrix and a vector $f(A,b)=y^\top (I-A)^{-1} b$ and I want to know the conditions under which it is convex. For functions of a vector, the positive definiteness of the Hessian ...
1
vote
1answer
22 views

Solve system of non-linear differential equations

I have a large system of non-linear differential equations of the form: $x_{i}''(y) = f_{i}(x_{1}, x_{2},\ldots,x_{n},x_{1}', x_{2}',\ldots,x_{n}', x_{1}'', ...
0
votes
0answers
26 views

Primal-dual Newton Barrier Method for Nonlinear Functions

I am writing the code for primal-dual Newton barrier method for nonlinear functions (algorithm described here in section 2). I have a few questions about the implementation details in the paper. The ...
1
vote
0answers
63 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 ...
1
vote
1answer
31 views

Example of maximal monotone operators in non-reflexive Banach spaces with applications in PDE

My question is about examples of maximal monotone operators that are defined in non-reflexive Banach spaces and have applications in PDEs, variational inequalities, etc (any application actually)? If ...
0
votes
0answers
36 views

L_1 norm optimization as a sequence of linear optimizations?

Does someone know of numerical methods to approximately solve ${\bf x_0} = \min_{\bf x}\{ \left\|\bf Mx - b\right\|_1\}$ by using some sequence of linear optimizations? Links or ideas are both ...
1
vote
1answer
36 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], ...
1
vote
0answers
22 views

Penalty methods in Unconstrained problems

Let me define $p : \mathbb{R}^n \to \mathbb{R}$ as a penalty function for the feasible set $\mathcal F$ of an equality and inequality constrained problem $(P)$ $$(P) : \text{min}\space f(x) \space ...
0
votes
0answers
18 views

Nonlinear least squares problems with binary variables

I want to solve the heat equation $T_t(x,t) = - L_x . T(x,t) + F(x,t)$ in an edge-weighted graph where $L_x = \sum_i x_i e_{ij}$ is weighted Laplacian matrix of the graph. Then I conclude to the ...
1
vote
0answers
17 views

Convexity Proof with constraints on the gradient

Consider a minimization problem $(P)$ : minimize $f(x)$ subject to $\delta_C(x) \leq 0$ Now assume that $\emptyset \neq C \subset \mathbb{R}^n$ is convex and let $f: \mathbb{R}^n \to \mathbb{R}$ be ...
0
votes
0answers
16 views

About the alternating optimization

The problem is defined as follows: $$ min_{A,B,C} f(A,B,C) $$ and the problem couldn't solve by gradient descent or close-form solution. Thus, the usual way is to use the alternating optimization: ...
1
vote
1answer
34 views

Convexity proof - can I get some pointers?

Prove that $C \subset \mathbb{R}^n$ is convex iff $\forall m \in \mathbb{N}$ and every set of $m$ points $\{x_1,...,x_m\} \subset C$ we have that $\sum_{i=1}^m \lambda_i x_i \in C$ Where ...
0
votes
0answers
13 views

How to solve a multidimensional stochastic optimization program with minimizing operators?

How to solve the following optimization program? $$ \max_{u_i^h,u_i^l,i=1,2}\Pi =E\big[a\min\{u_1^h,u_2^h, K_1^h,K_2^h\}+b\min\{u_1^h,u_2^l,K_1^h,K_2^l\}\\ ...
1
vote
1answer
48 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 ...
1
vote
1answer
39 views

Optimization of Frobenius Norm and Nuclear Norm

How to solve the following optimization problem, \begin{equation} \boldsymbol{\hat{x}} = argmin_{\boldsymbol{x}} \frac{1}{2} \| \boldsymbol{x - y} \|_F^2 + \lambda \| \boldsymbol{x} \|_{*} ...
1
vote
0answers
38 views

Difficulty in understanding a solution: Constraint minimization of sum of Non-symmetric matrices

I am trying to understand why there is significance difference in the performance of two proposed solutions. Original question (Constraint minimization of sum of Non-symmetric matrices) ...
0
votes
0answers
11 views

How to minimize $ \int_{\Omega} \frac1{2 \theta} (u(x) - v(x))^2 + \lambda|\rho(v(x))| dx $

I'm new in optimizations and i am trying to understand how to obtain $ v $ that minimizes $ \int_{\Omega} \frac1{2 \theta} (u(x) - v(x))^2 + \lambda|\rho(v(x))| dx $ where $\rho(x)$ - continuous ...