Questions tagged [nonlinear-optimization]

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

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8 views

Coordinate expansion or multiple regressions

I was wondering if anyone could offer some advice on the most productive direction to head in when seeking to fit a regresion on the below data which is most entirely centered on zero. Currently ...
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87 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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1answer
56 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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1answer
91 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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1answer
104 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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1answer
66 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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1answer
163 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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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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1answer
112 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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56 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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68 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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2answers
193 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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29 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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33 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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1answer
85 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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1answer
69 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-...
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50 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 ...
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1answer
54 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, ...
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37 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\}...
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1answer
77 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 ...
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1answer
81 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(...
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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 ...
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1answer
120 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)/...
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53 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=...
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1answer
212 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=...
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2answers
114 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 ...
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0answers
44 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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29 views

How to derive Lipschitz constant of Moreau envelope of a Lipschitz function

This question is from a lecture note of convex optimization. Q: Prove: If $f$ is $L$-Lipschitz, then its Moreau envelope $f_{\mu}$ is also $L$-Lipschitz. ($L = \mu^{-1}$) [NOT my homework] $$f_{\mu}(x)...
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0answers
38 views

A question about strong slopes (nonsmooth analysis)

Context. I'm reading the manuscrip "Nonlinear Error Bounds via a Change of Function" by Dominique Azé and Jean-Noël Corvellec (J Optim Theory Appl 2016), and I'm having a hard time ...
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45 views

Convex optimization under asymmetric loss in infinite dimensional space

The following problem is common in financial economics $$ \min_{m \in L^2} \mathbb{E}[ \phi(y(\theta)-m)] \quad \text{s.t. } \mathbb{E}[ mx ]= q $$ That is, given a random variable $y(\theta)$ ($\...
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0answers
18 views

Maximizing the volume of the intersection of a fixed ball with a cube with varying width and location

Given a ball $B$ and a linear subspace $L$ in $\mathbb{R}^n$, what is the maximum value of $\frac{vol(B \cap C)}{vol(C)}$ where $C$ is a cube of the form $x + [0, h]^n$ for $x \in L$ and $h \in \...
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0answers
26 views

Optimality of a simple solution of a linear fractional minimax problem

Consider the following linear fractional optimization problem \begin{align} \max_\mathbf{x}&\quad \min_{n=1,\ldots,N}\frac{x_n}{\alpha+\sum_{m}\beta_m^{(n)}x_m}\\ \text{subject to}&\quad\...
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0answers
15 views

Using Regula-Falsi to determine the solution to a non-linear system [closed]

Apologies, for this isn't a field or subject I know much about. Regula Falsi (I believe some may know this as "double false position" or something like this) can be used trivially, of course,...
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1answer
107 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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0answers
37 views

Prove that a polygon is convex over a circle

The problem Let $C_A$ (resp. $C_B$) a circle of center $A = (x_A,0)$ (resp. $B = (x_B,0)$) and radius $r_A$ (resp. $r_B$). For $k = 0,1,2,3,4$, let $D_k$ some points on $C_A$ with $D_0 = (x_0,0)$ Let $...
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0answers
46 views

Minimising kurtosis (non-convex). Can I use algebraic geometry or alternate methods to show uniqueness of a particular solution?

I consider a weighted sum of $n$ identically-distributed correlated random variables. The weights in the sum, $w_i$ for $i=1, 2,...,n$, satisfy $w_i>=0$ and $\sum_{i=1}^{n}w_i=1$. I am ...
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0answers
63 views

Sparse signal recovery (nonlinear case)

Let $K \subset \mathbb{R}^n$, it may be that $K$ is "very thin" (e.g. $K$ is a $k$-dimensional affine subset of $\mathbb{R}^n$, with $k \ll n$). I'm interested in the case where $K$ is ...
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37 views

How can I analyze the the effect of a constant on the arguments that minimize a function?

Background I have a function $J$ that I am minimizing, but the function is too expensive to minimize computationally. I derived an upper bound on $J$ (denoted by $\overline{J}$) that is not so hard to ...
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1answer
89 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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5answers
528 views

Elementary inhomogeneous inequality for three non-negative reals

I need the following estimate for something I am working on, but I don't immediately see how to establish it. For $x, y, z \in \mathbb{R}_{\ge 0}$, show that $$2xyz + x^2 + y^2 + z^2 + 1 \ge 2(xy + yz ...
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0answers
18 views

Dealing with degeneracy in nonlinear programming by “small” perturbations of constraints

CONTEXT: Suppose you have the nonlinear program $$ \begin{aligned} &\min f(x)\\ \text{subject to: }\quad & h_1(x) = 0 \\ &\quad\quad\vdots\\ &h_m(x) = 0 \end{aligned} $$ where $x\in\...
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1answer
270 views

An elementary inequality for three complex numbers

The following problem arose in asymptotic analysis of difference equations. Numerical maximization suggests that for all nonzero complex numbers $a,b,c$ we have $$h\big(r(a,b,c),r(b,c,a),r(c,a,b)\big)...
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30 views

Optimizing upper and lower bounds

Let $L_i:X\rightarrow [0,\infty)$ be continuous (objective) functions defined on a metric space $X$ and suppose that $$ L_1(x)\leq L_2(x)\leq L_3(x)\qquad (\forall x \in X). $$ Here, I imagine that $...
2
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1answer
93 views

Using Nelder-Mead to solve system of polynomial equations

I am trying to solve a system of $9$ polynomial equations in $9$ unknowns over the non-negative reals. Since the equations are quite large and I would like to use VBA, I prefer an algorithm that ...
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0answers
49 views

An optimization problem about number series

Given $m>0$, we want to minimize $$ \sum_{k=1}^r a_k \log b_k $$ for arbitrary increasing number series $a_k\geq 1$ and $b_k\geq 1$ satisfies $$ \sum_{k=1}^{\infty} \frac{1}{a_k}=1 $$ and $r$ ...
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2answers
373 views

Determining if a quadratic form is non-negative if variables are non-negative

Let $f(x_1,\dots,x_n) = \sum_{1 \le i \le j \le n} c_{i,j}x_ix_j$ be a homogeneous quadratic form. Is there a quick-ish way to determine whether $f(x_1,\dots,x_n) \ge 0$ for all $x_1,\dots,x_n \ge 0$? ...
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1answer
123 views

Dual problem with integrals

I am reading a paper where the author derives the following Lagrangian dual problem : $\min_v \int_R \frac{1}{4} \frac{\beta^2}{v-2\|x\|}dx+v\;\;\;\text{s.t.}\;\;\;v\geq 2\|x\|\;\;\;\forall x \in R$ ...
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0answers
48 views

Multiobjective Optimization with (too many?) functions

Consider a multiobjective optimization problem $$\min\limits_{x\in \Omega} f(x),$$ where $f:\Bbb R^n \rightarrow \Bbb R^m$ and $\Omega \subseteq \Bbb R^m.$ A point $\bar{x} \in \Omega$ is said to be: ...
3
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0answers
82 views

Convex optimization upper bound for a non-linear optimization

Is there any good convex optimization problem based upper-bound for the following non-linear optimization problem? \begin{align} \max_{x_1,\ldots,x_N}&\quad \sum_{n=1}^{N} \log(1+\frac{x_n}{1+\...
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1answer
116 views

Difference of two optimization problem's optimal value

Let we have two following optimization problems: \begin{align} \text{(P1)}\quad \alpha_1 = \max_{x_1,\ldots,x_M} &\quad \sum_{m=1}^{M}\log(1+f_m(x_1,\ldots,x_M))\\ \textrm{s.t.} &\quad \...

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