# Questions tagged [nonlinear-optimization]

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

498
questions

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### How to derive Slime Mould Equations

As shown in https://en.wikiversity.org/wiki/Slime_Mould_Algorithm, we have the Slime Mould equations for Approach food and Wrap food. I do not know how to derive these equations. Please help me or ...

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174 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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102 views

### Metric / strong slope restriction of function on unit ball in $\mathbb R^m$

Diclaimer. I'm not sure this is the right venue for this question, but I'll give it a try
Definition [Strong / metric slope]. Given a complete metric space $(M,d)$ and a function $f:M \to (-\infty,+\...

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50 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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97 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 ...

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26 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 ...

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162 views

### Is it possible to find an upper bound and a lower bound for $(f(\xi)-f(\tilde{\xi}))^T(\frac{\partial f}{\partial \xi}(\bar{\xi}))(\xi-\tilde{\xi})$?

For a system engineering problem I have to solve the problem below, but since I am not a mathematician, I am not sure if I have enough knowledge to solve it.
Problem definition: Let $f(\xi) \in \...

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42 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 ...

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97 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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330 views

### Limits of argmin ratios and sums

In my research on convergence properties of certain Bayesian methods I have encountered $\mathop{\mathrm{arg\,min}}$ limits of the forms
\begin{equation}
\lim_{n\to\infty} \mathop{\mathrm{arg\,min}}_{\...

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42 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 ...

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97 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.
...

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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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180 views

### Of all probability matrix $P$ having stationary distribution $\pi$, find the one having smallest diagonal

I am requesting your help today trying to solve a somewhat odd problem. Is there a way to find through some numerical algorithm such as Newton's method the stochastic matrix $\boldsymbol{P}$ having ...

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230 views

### A “nice” (but non-definite) quadratic programme

For integers $n\geq k>0$, let $f$ be the following quadratic form:
$$f(x_1,\ldots,x_n)=\sum_{i=1}^n\sum_{j=0}^{k-1}x_ix_{i+j\bmod n}.$$
Is it true that the minimum of $f$ over the unit simplex is ...

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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}{\...

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134 views

### Reference request on computational schemes for $\inf_{x\in\Omega^n}\sup_{y\in\mathbb R^n}F(x,y)$

Let $\Omega\subset \mathbb R^d$ be compact, $\rho$ be a density function on $\Omega$ and $p_1,\ldots, p_n\in (0,1)$ be weights satisfying $\int_{\Omega}\rho(z)dz=1=\sum_{k=1}^n p_k$. We consider the ...

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44 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 ...

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164 views

### How to sweep the leaves efficiently?

A cleaner, denoted by $P$, aims to sweep $n\ge 1$ leaves that appear one by one in a courtyard modeled by a compact set $D\subset \mathbb R^2$. Denote by $x_0$ the initial position of $P$ and by $v>...

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616 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 ...

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40 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 $f:V^2 \to \mathbb R$, $(...

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405 views

### Maximizing a function that is sum of gaussians

Let $\mathbf{x}_1,\dots,\mathbf{x}_n$ be given $n$ vectors in $\mathbb{R}^d$. Define the function
\begin{align}
\mathcal{K}(\mathbf{x},\mathbf{y})=
\alpha\exp(-\frac{||\mathbf{x}-\mathbf{y}||^2}{2\...

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105 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.}\;\;\;...

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79 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

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191 views

### optimization to find the maximum of sum of sigmoids with some constraints

I am having a problem of maximizing a sum of sigmoid functions over different time instants with some constraints.
Considering the standard sigmoid function $f(x)=\frac{1}{1+e^{-\alpha x}}$ and it's ...

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221 views

### Looking for a very particular kind of non-convex functions

I want some examples (hopefully parametric families!) of non-convex functions which satisfy the following properties simultaneously,
It should be at least twice differentiable.
It should have a ...

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203 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}^...

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203 views

### Hardness of concave minimization problem

I have an optimization problem $\underset{x}{\min} ~ c(x) - k \cdot x$ where $c(x)$ is a non-decreasing concave function with $c(0) = 0$, $x \in C \subseteq \mathbb{R}^d_{\geq 0}$. By non-decreasing, ...

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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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69 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) ...

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22 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
$$
(\nabla f(x) - \nabla f(y))^\top(x-y) \ge \mu \|x-y\|^2.$$...

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108 views

### How do I get an analytical solution to this nonlinear equation?

I posted this question over on Math Stack Exchange (link), but have not received a response. I'm wondering if it's too complicated for that audience, so I'm posting it here in the hopes that someone ...

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491 views

### Difference between Chebyshev first and second degree iterative methods

Consider linear equation $Au = f$.
We want to solve it with iterative method (assuming $A$ is good).
First order iterative method is:
$$
u^{k+1} = u^k - \alpha_{k+1}(Au^k - f),
$$
The second degree ...

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105 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-...

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106 views

### Fritz-John conditions: Equality-constrained case as special case of inequality constraints

In Chapter 4 of Nonlinear Programming: Theory and Algorithms by Bazarra, Sherali, and Shetty, the following claim is made after Theorem 4.3.2 (Fritz-John necessary conditions):
"Note also that these ...

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67 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 ...

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107 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 ...

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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_{...

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118 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 ...

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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=...

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1k views

### Minimizing product subject to linear constraints

I am looking for a solver that allows me to solve an optimization problem of the form
$$\begin{array}{ll} \text{minimize} & x_1 x_2 \cdots x_n\\ \text{subject to} & \color{gray}{\text{(some ...

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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.....

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62 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)...

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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 ...

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220 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)...

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110 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 ...

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240 views

### When is the optimum of an optimization problem affine in the constraint parameter?

While working on a variational problem I have reached to the following question:
Let $f:(0,\infty) \to [0,\infty)$ be a $C^1$ function satisfying $f(1)=0$. Suppose that $f(x)$ is strictly increasing ...

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32 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 ...

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60 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\...

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74 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-...