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Questions tagged [nonlinear-optimization]

Nonlinear objectives, nonlinear constraints, non-convex objective, non-convex feasible region.

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1answer
41 views

Does coercivity/supercoercivity conjugates?

According to Wikipedia, a function $f: \mathbb{R}^n \to \mathbb{R} \cup \{-\infty, +\infty\}$ is called coercive if, $$f(x) \to +\infty \text{ as } \|x\| \to +\infty$$ and it is super-coercive if $$\...
2
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0answers
41 views

Three-constraint homogeneous QCQP

Consider the homogeneous quadratically constrained quadratic program, $$\min_{u^T u =1} u^T A_1 u$$ $$\textrm{subject to}\quad u^T A_2 u = 0,\quad u^T A_3 u = 0$$ This problem is particularly studied ...
2
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0answers
42 views

Why not use global optimization algorithms like PSO to solve decentralized control problems?

I do not see many works that use global optimization algorithms to solve decentralized control problems. Here the decentralized control problem means some entries of the feedback matrix are ...
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0answers
21 views

How to find least-square point to plane alignment in SE(3)?

Given $\mathbf{l}_i\in\mathbb{R}^3$, $c_i\in\mathbb{R}$, and $\mathbf{x}_{ij}\in\mathbb{R}^3$, $i=1\cdots M$, $j=1\cdots N$, \begin{equation*} \begin{aligned} & \underset{R\in SO(3), \mathbf{t}\in\...
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0answers
258 views

The busy Star Guardian

On an infinite plane, the Prime Star has disintegrated into four constituent stars, the North Star, the South Star, the East Star and the West Star, each traveling at a constant speed of $1$ in their ...
2
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0answers
57 views

Analytical solution of 2nd order nonlinear ODE ($y''+(a+by^2)y'+cy = 0$)

I encountered the following ode in the attempt to solve the problem of nonlinear van der pol equation. I have tried for a long time to give it a solution but failed. $y''+(a+by^2)y'+cy = 0$ where $a$, ...
3
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1answer
224 views

Minimiser of a certain functional

Let $f_i \in L^1 ([0, 1])$ be a sequence of functions equibounded in $L^1$ norm - that is, there exists some $M > 0$ such that $\|f_i\|_{L^1} < M$. Define the functional $F: L^1([0, 1]) \to \...
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1answer
101 views

Was a quotient of two norms considered as a constraint to a convex optimization problem before?

I want to solve the optimization problem $$ \text{minimize }g(x) \quad \text{subject to} \quad \Vert x\Vert_{\infty}/\Vert x\Vert_{2} \le s $$ for $x\in\mathbb{R}^d$ and $s\in(0,\infty)$. The function ...
3
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1answer
67 views

Numerical scheme for convex optimization

Given $(e_n)_{-N\le n\le N}\in\mathbb R^{2N+1}$ and $-1<x<1$, solve \begin{eqnarray} &&\max_{(q_n)_{-N\le n\le N}\in\mathbb R^{2N+1}_+}~ \sum_{n=-N}^N (e_n-\log(q_n))q_n \\ \mbox{s.t.} &...
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0answers
27 views

Breaking up an infinite-dimensional optimization problem into a sequence of finite-dimensional problems

My question is a bit vague. I have an infinite-dimensional convex optimization problem and I can solve constrained versions of the problem by restricting the domain of the objective function to a ...
0
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0answers
55 views

Chain rule for Clarke's subdifferential

My question is on the validity of the chain rule for Clarke's subdifferential for functions into spaces of dimension larger than $1$. In Clarke's original work (Optimization and nonsmooth analysis) as ...
0
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0answers
43 views

Nested, successive minimization solved by asympotic minimization?

I am curious about the general relation between nested, successive minimization (M1) and asymptotic minimization (M2) as defined in the following. What one wants is to implicitly minimize a sequence ...
4
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0answers
103 views

An inequality for three iid random variables with a log-concave density

It was previously shown that $$H\ge cG,\tag{1}$$ where $c:=1/14334$, $$G:=E|X-Y|,\quad H:=E|X-Y|-\tfrac12\,E|X+Y-2Z|,$$ and $X,Y,Z$ are independent random variables with the same log-concave density. ...
2
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0answers
99 views

Existence and uniqueness of solution of a nonlinear system

I need a proof of the following result to calculate a Nash equilibrium in the Showcase Showdown game. For all $n>1$, the system of equations $$\left\{ \begin{aligned} (1+e^{x}(-1+x))^{n-2}&=\...
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0answers
31 views

Lipschitz solutions to linear complementarity problems (LCP)

Let $M\in\mathbb{R}^{n\times n}$. For $q\in\mathbb{R}^n$, define the set: $$S_M(q)=\{y\in\mathbb{R}^n|y\ge 0,q+My\ge 0, y^\top (q+My)=0\}.$$ This is the set of solutions to the LCP $(q,M)$. We say $...
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0answers
44 views

Iterative minimization of a functional $f=f((x(\xi^1,\xi^2, ...), \frac{\partial x}{\partial \xi^1 }, \frac{\partial x}{\partial \xi^2 }, ...)$

Consider the minimization of a nonlinear functional $f$ of a field $x$ and its partial derivatives, $$ f=f\left(x(\xi^1,\xi^2,\dots,\xi^n), \frac{\partial x}{\partial \xi^1 }, \frac{\partial x}{\...
2
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0answers
53 views

Continuum of Lagrange multipliers, duality gap, and minimax theorem

Suppose I have a linear optimization problem involving random variables on some (infinite) probability space $\Omega$. For example, need to maximize expectation $E[Q]$ of random variable $Q$ subject ...
5
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2answers
675 views

What is Young measure?

I read about Young measures from the book, Weak convergence methods for nonlinear partial differential equations by L.C. Evans. He introduces the concept by the following theorem: Theorem. Assume ...
1
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0answers
43 views

Minimize smooth function $(x,y) \to f(x,y)$ subject to $x \perp y$

Let $V$ be a finite-dimensional real vector space (e.g space of $m \times n$ real matrices equiped with Hilbert-Schmidt inner product $(A,B) \to \mathrm{tr}(AB^\top)$, and let $f:V^2 \to \mathbb R$, $(...
2
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0answers
107 views

Optimization of functionals with constraints

I have a minimization problem as follows: $\min\left( \int_0^1\int_0^1\beta(t)\beta(s)G_1(t, s)dtds\right)^{1/2}+\left( \int_0^1\int_0^1\beta(t)\beta(s)G_2(t, s)dtds\right)^{1/2} $ $\texttt{s.t.}\;\;\;...
1
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2answers
81 views

Variant of Parthasarathy's minimax theorem

Does there exist a variant of Parthasarathy's minimax theorem [1] that relaxes the assumption that the spaces $X$ and $Y$ are $[0,1]$? [1] https://en.wikipedia.org/wiki/Parthasarathy%27s_theorem
3
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1answer
210 views

Maximizing the distance sum of some points inside a circle

Consider $n$ points $\{p_i\}_{i=1}^n$ located inside or on a circle with radius $r$ in the plane. The question is: how to place the $n$ points so that the sum of inter-point distances, $$J=\sum_{i=1}^...
3
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1answer
254 views

Eigenvectors of a tensor in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$

I want to find the critical point of tensor $f=a_0b_0c_0 + a_1b_1c_1$ in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$, and I followed this construction: First, I take the following partial ...
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0answers
24 views

Are such assumptions of functions similar to strong convexity reasonable in convex optimization?

For $\mu$-strongly convex function $f:\mathbb{R}^d\to\mathbb{R}$, the following property holds: for any given $x,y\in\mathbb{R}^d$, we have $$ (\nabla f(x) - \nabla f(y))^\top(x-y) \ge \mu \|x-y\|^2.$$...
2
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0answers
119 views

Condition under which the Clarke's subdifferential is locally Lipschitzian

Given a locally Lipschitz continuous function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ and closed set $$\Omega =\left\lbrace x \in\mathbb{R}^n \ |\ f(x) \leq 0 \right\rbrace$$ such that f is semi-...
4
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1answer
70 views

Do Pareto critical points of a multicriteria optimization problem form an attractor of the dynamical system induced by a descent algorithm?

Let $d\in\mathbb N$, $k\in\mathbb N$ and $f:\mathbb R^d\to\mathbb R^k$ be differentiable. Say that $v\in\mathbb R^d$ is a descent direction at $x\in\mathbb R^d$ if ${\rm D}f(x)v<0$ (component-wise) ...
1
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1answer
96 views

Calculus of variations for double sum with Lagrange multiplier

This cropped up in a research question I'm tackling. I wish to solve the following optimization problem: $$ \text{minimize}\ \sum_{i=1}^\infty f_i \sum_{j=1}^i \sqrt{f_j} \quad\text{subject to}\ \sum_{...
0
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1answer
108 views

What is the computational complexity of the calculation of $ \Psi(x) $?

What is the computational complexity of the calculation of $ \Psi(x) $ described below: Let $\left\{ f_i : \{0,1,\dots,m\} \to \mathbb{R} \right\}_{i=1}^n$. For each $x \in \{0,1,\dots,m\}$ we ...
1
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2answers
109 views

Does this non-negative function, with No stationary points, have only descend directions close to a constraint set?

Suppose $P: \mathbb{R}^n \rightarrow \mathbb{R}_{\ge 0} $ is differentiable map, with $P(x) = 0 \ \forall x \in \mathcal{X}$ and $P(x) > 0 \ \forall x \in \mathcal{X}^c$. Further, suppose $P$ has ...
1
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1answer
166 views

How to solve a system of nonlinear equation, with y known and x or its coefficients unknown? [closed]

While solving a complex problem I have ended up with this simplified problem: There are eight straight lines in the plane. They are notated as follows: \begin{gather} \tag{1} \label{1} y=k_1 x+b_1\\ y=...
0
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0answers
25 views

Norm of vector components optimization of linear matrix combination

Given complex matrices $A_1, A_2, \dots, A_k\in\mathbb{C}^{m \times n}$, $B \in\mathbb{R}^{m \times n}$, the objective is to find a vector $x \in \mathbb{C}^k$ such that: $\max {||x_i||}$ , $i\in 1,2.....
0
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1answer
119 views

What to call a function that is negative on a set

Let $Y$ be a nonempty region in $\mathbb{R}^n$. I am designing an algorithm which given a point $x_0$ outside $Y$ in a finite number of steps lead to a point $x_n∈ Y$. The way I do it is that I have a ...
2
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0answers
65 views

Monotone rearrangements of function, constrained optimization in $\mathcal{L}^p$

By $\mathcal{S}$ let us denote the set of such step functions $f:[0,1]\to [0,1]$ that additionally satisfy: $$\forall_{ x>\frac{1}{2}} \ \ \lambda\Big(f=x\Big) \ = \ x\cdot \Big[\lambda\Big(f=x\Big)...
4
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0answers
74 views

Minimizing the largest eigenvalue of matrix product

Let $A\in\mathbb{C}^{m\times n}$, $B\in\mathbb{C}^{n\times k}$, $C\in\mathbb{C}^{k\times m}$ be given complex matrices. The objective of the optimization problem is \begin{equation} \mathop {\arg \min ...
3
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2answers
233 views

On some inequality (upper bound) on a function of two variables

There is a problem (of physical origin) which needs an analytical solution or a hint. Let us consider the following real-valued function of two variables $y (t,a) = 4 \left(1 + \frac{t}{x(t,a)}\right)...
0
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0answers
33 views

Numerically finding matrix approximation by lower-dimensional "pseudo-similar" matrix

Consider an $N\times N$ (real or complex) matrix $A$, and some $n<N$. Is there a good numerical algorithm that finds the set consisting of an $n\times n$ matrix $B$, an $n\times N$ matrix $I$, and ...
2
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0answers
87 views

Global minimum of sum of a non-convex and convex function, where minima of the non-convex function can be found

I'm interested in finding $\arg\min_{x \in X} (f(x) + \lVert x\rVert_2^2)$ where $X$ is a $[0,1]^n$, $f$ is Lipschitz but non-convex and we already have a procedure to find some $x^* \in \arg\min_{x\...
1
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1answer
134 views

Example of a differentiable function optimization where derivative free methods are used

While preparing a workshop on the derivative free methods, and fminsearch in MATLAB, I found an example function where fminsearch converges better and in less iterations than fmincon with calculated ...
0
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1answer
83 views

Gradient-descent "type" Methods for non-convex and non-smooth functions

Most (stochastic) "gradient descent" type algorithms (such as Nesterov-accelerated gradient-descent or ADAM) seem to be well-defined only for functions which are either: lower semi-...
1
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0answers
60 views

Taut string algorithm and TV-minimization equivalence

Given real numbers $y_i's$, consider the following convex optimization problem: $$ \min_{x_i's} \sum_{i=1}^N(y_i-x_i)^2 + \lambda\sum_{i=1}^{N-1}|x_{i+1}-x_{i}|. $$ The paper A Direct Algorithm for 1D ...
-1
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1answer
59 views

Existence of continuous selection for metric projection

Let $(X,d)$ be a separable complete geodesic metric space and let $K$ be a compact (non-empty) subset of $X$. Without assuming things like linearity, the convexity of $K$, and locally convexity, ...
0
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0answers
40 views

A parametrized saddle point problem with linear constraints

I am struggling to find any potential algorithm for solving a saddle point problem. More precisely let $\mathcal{P}=\{ \mathbf{x}\in \mathbb{R}^{d}; \mathbf{A}\mathbf{x}=\mathbf{b}, \mathbf{x} \geq 0\}...
2
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1answer
114 views

Normal cones and KKT conditions

I'm trying to understand a statement from the book "Perturbation Analysis of Optimization Problems", by Bonnans and Shapiro. Let me start by providing some context. In page 148, the authors ...
3
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1answer
116 views

KKT conditions of problem with variational inequality constraint

I have an optimization problem with a variational inequality constraint: $$ \begin{equation} \begin{array}{ll} \min_x & f(x) \\ \mathrm{s.t.} & g_i(x) \leq 0, \quad i=1,\ldots,m \\ & h_j(...
0
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0answers
20 views

Distributed optimization - expectation of a product

I've been trying to find distributed optimization algorithms for solving a problem of the form: $$ \min_x \mathbb{E}\left[f_1(x) \cdot f_2(x) \cdot \ldots \cdot f_N(x)\right], $$ where each agent only ...
4
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1answer
134 views

Nonlinear system of integral equations

I have encountered a system of nonlinear integral equations in my work. They take the form $$\int_{0}^{1} \frac{1}{g(y)}e^{f(x)/g(y)}(x+f(x)/g(y)-f(x))dy=0$$ $$\int_{0}^{1}\frac{f(x)}{g(y)^2} e^{f(x)/...
2
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0answers
56 views

Minimizing a certain norm of the identity operator on $\mathbb R^2$

$\newcommand\R{\mathbb R}\newcommand\Q{\mathcal Q}$For mutually orthogonal vectors unit vectors $a=[a_1,\dots,a_n]^T$ and $b=[b_1,\dots,b_n]^T$ in $\R^n=\R^{n\times1}$ (so that $n\ge2$) and for all $x=...
5
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1answer
226 views

On a certain norm of the identity operator on $\mathbb R^2$

$\newcommand\R{\mathbb R}\newcommand\Q{\mathcal Q}$For mutually orthogonal vectors unit vectors $a=[a_1,\dots,a_n]^T$ and $b=[b_1,\dots,b_n]^T$ in $\R^n=\R^{n\times1}$ (so that $n\ge2$) and for all $x=...
1
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2answers
119 views

Robust estimation of $Ax=b$

Problem setting : $ \underset{x}{\text{min}} \|Ax-b\|$, where $A \in \mathcal{R}^{m \times n}, m\gg n $, full rank. L1 loss is used for robust estimation using IRLS. The corresponding equation to ...
2
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0answers
46 views

Which algorithm to optimize this problem?

I do need to find coefficients of a parametric model given observations, and I was wondering which algorithm I should use. The problem is as follows. I have a set of values $\mathbf x_i = (x_{i,1},\...

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