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
20 views

Can you suggest a method to split a function in two parts such that the two parts may represent equitable magnitude contribution.

I was thinking about the sum of a even and odd function but the domain for the problem is positive real numbers only.
0
votes
0answers
17 views

SOCP or SDP optimization problem

I am studying an optimization problem \begin{equation} \mathbf{w}^* = \text{argmax} \sum_{d=1}^D \log \bigg( \frac{|\mathbf{f}_d^H\mathbf{w}|^2+c_1}{|\mathbf{f}_d^H\mathbf{w}|^2+c_2} \bigg)\\ \\ ...
2
votes
2answers
28 views

Sensitivity of polynomial global minimizers with respect to perturbations in the coefficients.

I'm trying to find the value of a global minimizers of a multivariate polynomial (4 variables) of high order numerically. The numerical values of the coefficients are coming from noisy measurements ...
0
votes
0answers
47 views

Which is the better way to optimize a function with 3 variables

I have an optimization function depends on 3 parameters a, b, and c. Which is the better way to optimize it? ...
0
votes
0answers
30 views

Newton's method for unconstrained optimization applied to a quartic function in R2

I am faced with the task of applying Newton's method to the following problem: $$ \text{min} ~~~~~ 8x_1x_2+\frac{1}{4}(x_1-x_2)^4 $$ where $x \in \mathbb{R}^2$. For clarification, the Newton method ...
0
votes
0answers
13 views

slaters condition - Duality - KKT condition [closed]

Can someone give a more intuitive idea to Slater's conditions and how it is related to KKT condition and duality ?
2
votes
1answer
12 views

About constrained optimization

I've the following optimization problem:$$\min f(\theta_1,\theta_2)=\frac{a}{\cos\theta_1\cdot v_1}+\frac{b}{\cos\theta_2\cdot v_2}$$$$\operatorname{sub}\quad a\cdot\tan\theta_1+b\cdot\tan\theta_2=c$$ ...
2
votes
1answer
34 views

Why is the conjugate direction better than the negative of gradient, when minimizing a function

In gradient descent we minimize a function $f(\textbf{x})$, by using the update rule: $$\textbf{x}_{t+1} = \textbf{x}_t-\alpha\nabla f(\textbf{x}_t).$$ We also know, that at each iteration we have ...
1
vote
1answer
16 views

A weird optimization problem

I've the following optimization problem:$$\max f(R,z)=R^2(a+z)$$$$\operatorname{sub}\begin{cases}R^2+z^2=a^2\\0\le z \le a\end{cases}$$ Once solved it gives $z=a/3$, ... Consider now the ...
0
votes
0answers
38 views

Can a positive definite kernel produce a kernel matrix which has negative eigenvalues?

(1) I've read that a symmetric matrix is positive definite when its associated eigenvalues are all positive. I am learning SVM lately, and have come to know a $d$th-degree polynomial kernel ...
0
votes
1answer
37 views

What is the correct change of variables to yield convexity in this nonlinear optimization problem?

$$ \text{min. } x/y \\ \text{s.t. } 2\leq x \leq 3 \\ x^2+y/z\leq \sqrt{y} \\ x/y=z^2 \\ x,y,z\geq 0 $$ To transform this problem into a nonlinear convex optimization problem, both the objective ...
1
vote
1answer
34 views

Why test problems in convex optimization are mostly random?

Very often people who compare performance of different algorithms in convex optimization use randomly generated data. For instance, this often happens in compressed sensing and signal processing. Is ...
0
votes
0answers
28 views

Optimization problem in image processing

I recently heard of the interior point method in nonlinear optimization problems and was also told of its (likely) usefulness in image processing. MATLAB of course, has just a function for that called ...
0
votes
0answers
15 views

Linearization of non-linear model

How can I linearize this: $b{\rm{log_2}}(1+xy)$ where $b\in\{0,1\}$: binary integer variable $0\le x\le 3$: continuous variable (bounded) $y$ is a known constant greater than $0$.
0
votes
0answers
10 views

How to perform the linearization

Can anyone help me to linearize this following constraint: $b_1{\rm{log_2}}(1+x_1y_1)+b_2{\rm{log_2}}(1+x_2y_2)\le Az$ Here $b_1$ and $b_2$ are binary integer variables. $0\le\{x_1,x_2\}\le3$: ...
0
votes
0answers
32 views

Solve: tanh(x) = a*x + b - most efficient way

I work on DSP code, where some equations are of form: tanh(x) = a*x + b (tanh or other hyperbolic functions) Currently I use Newton-Raphson method. Is there a better/faster method of finding ...
0
votes
1answer
26 views

How to linearize this mixed-integer nonlinear constraint

Can someone please help me to linearize the following nonlinear/nonconvex constraint: $\sum\limits_{n=1}^Na_n\rm{log_2}(1+x_ny_n)\le M\delta$ Here $a_n \in\{0,1\}$, binary integer variable $0\le ...
0
votes
1answer
33 views

How to linearise this nonlinear constraint

I want to linearize or convexify this following constraint. Here $c_t$ is binary integer variables, $p_t$ are continuous variable which are bounded. $\gamma$ is a continuous variable. $h_t$ and $V$ ...
0
votes
0answers
26 views

What is a good optimization algorithm/tool for otimization on Partially Ordered set?

Actually I'm interested to minimize following kind of functions: $f: U \rightarrow V$ where: $U$ is a vector space and $V$ is a Ordered vector space, i mean Partially Ordered Vector space. ...
0
votes
0answers
17 views

Solving large non-linear polynomial equation system

I have a 2 order equation system of 7 unknowns. It is constructed as this: F1=0,F2=0,F3=0...F7=0 of which F1=f1*f2,F2=f3*f4... And f1=a1*p1+a2*p2+a3*p3+a4*p4+a5*p5+a6*p6+a7*p7 a1~a7 are known ...
0
votes
0answers
33 views

How to convexify a function

I encountered a problem in nonlinear programming, the model is written: max a s.t. a = bd + c(1-d) where a, b, c, d are positive variables. b and c are bounded, and 0<=d<=1. I am wondering if ...
0
votes
1answer
34 views

Non linear programming

Could you please help me in solving the problem posted below. A company uses a raw material to produce two types of products. When processed, each unit of raw material yields 2 units of product 1 and ...
0
votes
0answers
15 views

Linear independence of equality constraint gradients in constraint qualifications

I'm, trying to get an intuitive feel for the various constraint qualifications for KKT points. Most of them seem to rely on the linear independence of $\nabla g_i(x^*)$ where $g_i$ are the equality ...
1
vote
0answers
21 views

Reduce degree of a high degree unconstrained binary term to quadratic unconstrained binary term

I'm working on a optimization project, in this project I have to convert higher order unconstrained binary polynomial to quadratic unconstrained binary polynomial. Can anyone give me a hint of how to ...
1
vote
0answers
49 views

Mixed-Integer Linear Programing : get the maximum constant associated to a non null variable

Does anyone know a way get the maximum constant associated to a non null variable using Mixed-Integer Linear Programing ? I would like to get the variable $a$ in this description : $$ i = 1,\ldots,m ...
1
vote
0answers
15 views

Can this specific Linear Program constraint be expressed? [duplicate]

Thanks for your time. I have a linear program and no idea how I could express a form of constraint and even if it's possible. Maybe someone here know a solution. A company assembly and sells a ...
1
vote
0answers
29 views

Line search Armijo, Wolfe, Strong Wolfe and Goldstein.

What are the articles (References) who proposed the line search of Armijo, Wolfe, Strong Wolfe, and Goldstein? Articles precursors of unidirectional searches?
0
votes
0answers
17 views

In the problem below, what is the right mix of drugs to maximize the expected revenue without exceeding R&D resources?

This is not a homework question. It's a question for a class I have yet to take that my friend gave me. I haven't been able to figure it out. Help is appreciated because it's driving me crazy. A ...
2
votes
0answers
33 views

Exercice the converge $r$-superlinearly

Give $x_0 \in \mathbb{R}^*$. Show that $\{x_k\} \subset \mathbb{R}$ converge $r$-superlinearly for $x^∗=0$, where $x_k$ is defined by $x_{k+1}=(1−\beta_k)x_k$ and $\beta_k=1−2^{-k}$ if $k=i^2$ for ...
1
vote
0answers
45 views

Proving/deciding concavity of a function of two variables

I would like to formally prove that the function $f(x,y) = \frac{(c+1)e^{-x}(xe^{x+y}+y)}{(c+2)(e^{x+y}-1)+e^y} $ is concave ($ c>2$ is a constant, and both $x,\, y \in \mathbf{R_+}$). Plots of ...
0
votes
0answers
17 views

gradient descent - cost reduces and then increases

I am optimizing a function using Gradient Descent. The learning rate is fixed. First for few iterations the cost decreases after that it starts increases. What is the reason for this?
0
votes
0answers
42 views

optimization of nasty expression with nice symmetry between expressions

Consider the function $\ f(x,y,z,\rho_a,\rho_b)=$ $ \log \left(1+ (x+ \rho_ay)^2 + \frac{(z+ \rho_by)^2}{1+(x+ \rho_by)^2} \right)+ \log\left(1+ (x+ \rho_by)^2 + \frac{(z+ \rho_ay)^2}{1+(x+ ...
4
votes
1answer
133 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: } \ ...
1
vote
0answers
29 views

When result of max of min problem is equal to min of max problem

Let's assume there are two functions $f(x)$ and $g(x)$. I want to know when the optimal $x$ of max of min of $f(x)$ and $g(x)$ is not equal to optimal $x$ of min of max of $\frac{1}{f(x)}$ and ...
1
vote
2answers
44 views

Is the set of all projection matrices a convex set?

The set $\phi=\{P| P^2=P\}$ contains all projection matrix. Is this set $\phi$ convex?
0
votes
0answers
14 views

Sequential Quadratic Programming

I am new to optimization techniques. I have lograthmic function which I need to maximize. I need to ask if sequential quadratic programming can be used to solve logrithmic function taken as objective ...
0
votes
1answer
25 views

Solving coupled non-linear equations

I am struggling to understand what the following question requires me to do: I believe I need to differentiate implicitly, but am unsure how I show it cannot be done.
1
vote
0answers
19 views

Explain KKT conditions without reference to duality.

Is it possible to explain (not derive) KKT necessary conditions without reference to the concept of Lagrangian duality?
0
votes
0answers
27 views

Constrained non-linear optimization problem

For some background, this comes from a sample size allocation problem in statistics. I am trying to minimize the following function (a sum of three variances), and could use some help with direction ...
0
votes
0answers
34 views

Saddle point problem (KKT) with block-diagonal matrix

Consider the following saddle point problem originating from an interior-point method algorithm: $$ \begin{bmatrix}\mathbf{H} & \mathbf{A}^{T}\\ \mathbf{A} & \mathbf{0} ...
0
votes
0answers
17 views

Non linear Programming Problem

I am struggling with the following question: Solve the following programing problem: max $f(x_1,x_2)$= $ \sqrt{(x_1 + 1) (x_2+1)} $ subject to $x_2-(x_1-1)^2 \leq 0 $; $x_1+x_2 \leq 7 $; $x_1, x_2 ...
0
votes
0answers
20 views

Separate a list of spheres into several lists, each contained in a sphere with a radius no larger than specified.

I have a list of arbitrary spheres, what I want to end up with is that list separated into a number of groups, where spheres in each group all fit into thier specific larger sphere. The limitation is, ...
0
votes
0answers
37 views

Fenchel dual vs Lagrange dual

Consider the Fenchel dual and the Lagrangian dual. Are these duals equivalent? In other words, is using one of the these duals (say for solving an optimization), would give the same answer as using ...
1
vote
1answer
36 views

Question regarding KKT conditions in optimization

Following is Proposition 3.3.7 in Bersekas' Nonlinear Programming. Let $x^*$ be the local minimum of the problem: $$\text{Minimize }\; f(x) $$ $$ \text{subject to: }\ h_j(x) = 0, ...
7
votes
0answers
69 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) ...
1
vote
0answers
21 views

Solving many independent non-linear systems simultaneously

I'm working on solving lots of systems of nonlinear equations. Luckily, the non-linear equation is the same, but the parameters are different: $$ f(\vec{x}_0; c_0) = 0\\ f(\vec{x}_1; c_1) = 0\\ ...
2
votes
1answer
56 views

min : sum of L2 norm and squared-L2 norm.

Is there a closed-form solution of the following convex problem: $$\min_x \| x - u \| + C \| x - v \|^2$$ where $\| \cdot \|$ is the L2 norm.
1
vote
2answers
26 views

How to mathematically prove the optimality conditions for a univariate function?

Consider a univariate function $f(x)$. I know the graphical intuition behind why $f'(x)=0$ at the extrema of $f$. But how do you prove it mathematically? I start with the assumption of $x^*$ being a ...
1
vote
1answer
38 views

Why don't we go beyond the Hessian in multivariate optimization?

In univariate optimization, we perform the first derivative test to identify stationary points and the second derivative test to classify the stationary points as minima, maxima and inconclusive. When ...
1
vote
1answer
53 views

Constrained Optimization : Minimize sum of dot products

I am working on a problem to minimize sum of dot product. The problem can be stated as following. Given a matrix where each element is either 0 or 1. $$ \ A_{ij} = \{0,1\}; $$ with the constraint ...