# Questions tagged [nonlinear-optimization]

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

504
questions

**0**

votes

**0**answers

21 views

### Is this gradient-descent-like algorithm which creates sparity

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a class $C^{1,1}$-function, $\lambda\geq 0$ be a ''learning rate'', $\lambda\geq 0$ be some "sparity generating parameter", $T\in \mathbb{N}$ be ...

**0**

votes

**0**answers

13 views

### Lagrange Optimization w.r.t. vector + equality constraint

I am currently facing the following optimization problem:
$f(v) = r_f + v'*\mu + v'*diag(\Sigma)*1/2 -v'*\Sigma * v*1/2+(1-\gamma)*v'*\Sigma*v*1/2$
s.t.
$1'*v=1$ (sum of the vector elements = 1)
...

**0**

votes

**1**answer

17 views

### Transformation of an unconstrained binary quadratic optimization problem into a constrained binary linear programming problem

I know that a constrained linear optimization problem can be transformed into an unconstrained binary quadratic optimization problem (UBQP). Does anyone know if the inverse result is solved in the ...

**0**

votes

**0**answers

23 views

### What's "Arrow-Hurwicz method" for solving saddle point optimization problems?

I have seen some papers on convex-concave optimization citing the "Arrow-Hurwicz method" from the paper [1] in different ways. However, since I cannot find a pdf version of this paper and ...

**1**

vote

**1**answer

35 views

### Relaxations for the spectral norm maximization problem

Optimizing the spectral norm of some positive semidefinite matrix $A(x) \in S^{n}$, w.r.t. a list of variables $x \in \mathbb{R}^d$ and semidefinite constraints is, in general, a nonconvex problem (...

**0**

votes

**0**answers

81 views

### Optimization problem where the objective function returns a function instead of a real number

As we all know, a classic optimization problem can be represented in the following way:
Given: a function $f: A \rightarrow \mathbb{R}$ from some set $A$ to the real numbers
Sought: an element $x_0 ∈ ...

**0**

votes

**0**answers

47 views

### Convergence of steepest descent in the nonquadratic case - formal proof of Theorem 3.4 in Nocedal & Wright's book

On page 43 of Nocedal & Wright's Numerical Optimization, the authors provide the following Theorem 3.4 without any proof:
Suppose that $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is twice
continuously ...

**0**

votes

**0**answers

59 views

### Calculus of variations for fractions of integrals

I would like to know if optimisation problems of the following form have already been studied: Let $\mathcal{F}$ be a class of functions $f:\mathbb{R} \rightarrow \mathbb{R}$. Minimize the following ...

**1**

vote

**0**answers

52 views

### Minimizing square roots with the consecutive ones property

Let $A=[a_{ik}]$ be a matrix with the consecutive ones property in each column, i.e. each column consists of a single consecutive block of $1$'s (with zeros everywhere else). Is there anything at all ...

**1**

vote

**0**answers

120 views

### Factorization of argmax

We consider a function $f(s_{1:p}, a_{1:p})$, where $p>1$ is an integer, $s_{1:p}$ denotes $(s_1,\ldots,s_p)^\top \in R^p$, and $a_{1:p}$ denotes $(a_1,\ldots,a_p)^\top \in R^p$.
Question: What is ...

**1**

vote

**1**answer

50 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**

votes

**0**answers

48 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**

votes

**0**answers

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 ...

**0**

votes

**0**answers

22 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\...

**8**

votes

**0**answers

264 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**

votes

**0**answers

60 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**

votes

**1**answer

231 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 \...

**1**

vote

**1**answer

110 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**

votes

**1**answer

71 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.} &...

**0**

votes

**0**answers

28 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**

votes

**0**answers

108 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**

votes

**0**answers

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**

votes

**0**answers

105 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**

votes

**0**answers

103 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}&=\...

**0**

votes

**0**answers

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 $...

**0**

votes

**0**answers

45 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**

votes

**0**answers

79 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**

votes

**2**answers

725 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**

vote

**0**answers

45 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**

votes

**0**answers

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**

vote

**2**answers

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**

votes

**1**answer

221 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**

votes

**1**answer

256 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 ...

**1**

vote

**0**answers

25 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**

votes

**0**answers

129 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**

votes

**1**answer

73 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**

vote

**1**answer

100 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**

votes

**1**answer

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**

vote

**2**answers

111 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**

vote

**1**answer

167 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**

votes

**0**answers

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**

votes

**1**answer

122 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**

votes

**0**answers

66 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**

votes

**0**answers

77 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**

votes

**2**answers

239 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**

votes

**0**answers

37 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**

votes

**0**answers

128 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**

vote

**1**answer

166 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**

votes

**1**answer

116 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**

vote

**0**answers

71 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 ...