Questions tagged [nonlinear-optimization]

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

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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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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 ...
José María Grau Ribas's user avatar
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Does this non-negative function, with no stationary points, have only descent directions close to a constraint set?

Suppose $P: \mathbb{R}^n \rightarrow \mathbb{R}_{\ge 0} $ is a 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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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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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 ...
Slava Rychkov's user avatar
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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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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 ...
hichem hb's user avatar
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3 answers
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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)...
Vladimir's user avatar
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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 ...
Andi Bauer's user avatar
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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\...
Proof by wine's user avatar
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1 answer
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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 ...
Andjela Todorovic's user avatar
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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-...
ABIM's user avatar
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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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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, ...
Catologist_who_flies_on_Monday's user avatar
2 votes
1 answer
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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 ...
Ariel Serranoni's user avatar
3 votes
1 answer
293 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(...
Daniel Turizo's user avatar
4 votes
1 answer
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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)/...
mwalth's user avatar
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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=...
Iosif Pinelis's user avatar
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1 answer
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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=...
Iosif Pinelis's user avatar
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2 answers
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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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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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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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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 $y(\theta)$ ($\...
Dejan Evisal's user avatar
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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 \...
pinaki's user avatar
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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,...
10GeV's user avatar
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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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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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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 ...
Brian's user avatar
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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 ...
Sébastien Loisel's user avatar
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1 answer
185 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,+\...
dohmatob's user avatar
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5 answers
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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 ...
BPN's user avatar
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1 answer
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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)...
Iosif Pinelis's user avatar
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0 answers
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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 $...
ABIM's user avatar
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2 votes
1 answer
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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 ...
Zorg's user avatar
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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 $r$ ...
gondolf's user avatar
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7 votes
2 answers
565 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 $x_1,\dots,x_n \ge 0$? ...
mathworker21's user avatar
1 vote
1 answer
297 views

Dual problem with integrals

I am reading a paper where the author derives the following Lagrangian dual problem : $\min_v \int_R \frac{1}{4} \frac{\beta^2}{v-2\|x\|}dx+v\;\;\;\text{s.t.}\;\;\;v\geq 2\|x\|\;\;\;\forall x \in R$ ...
OmarR's user avatar
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3 votes
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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+\...
Math_Y's user avatar
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1 answer
148 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 \...
Math_Y's user avatar
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Various definitions of coercivity

In this post one says that a functional $F:H\rightarrow [0,\infty]$ on an infinite-dimensional Hilbert space $H$ is (strongly) coercive if there exists a constant $k>0$ such that $$ F(x)\geq k\|x\|...
ABIM's user avatar
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2 votes
1 answer
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Lower semi-continuity of length-dependent functional

Let $f:\mathbb{R}\rightarrow [0,\infty]$ be a lower semi-continuous function and define the functional $$ \begin{aligned} F_f:&\ell^1 \rightarrow [0,\infty]\\ (x_n)_{n=0}^{\infty} &\to \sum_{n=...
ABIM's user avatar
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2 votes
0 answers
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Numerical algorithms for geodesically convex optimization

I want to solve a minimization problem of the form $\inf_{x \in M} f(x)$ where $M$ is a Hadamard manifold and $f$ is geodesically convex (but not differentiable). Since I know that in general a ...
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0 votes
1 answer
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Optimization problem involving matrix

I am struggling to solve an optimization problem of the following form: $$\begin{array}{ll} \underset{A}{\text{maximize}} & \log \det (A) \\ \text{subject to} & a^T A^{-1} a \le b\end{array}$$ ...
user164237's user avatar
2 votes
0 answers
171 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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50 views

What transformation is required to find a unique solution of this problem instead of multiple solutions?

$$ \max\limits_{\mathbf{f},\ \|\mathbf f\|=1 } \log_2\left(\prod^K_{i=1} \ \frac{ \mathbf{f}^H {\mathbf E} (\mathbf{W}_i, \Theta, \tau_i) \mathbf{f}} { \mathbf{f}^H \mathbf{G}_i ( \mathbf{W}_i, \...
syam's user avatar
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4 votes
1 answer
153 views

Is there a point in 6-dimensional space satisfying these polynomial inequalities?

I would like to know if there is a point $(a, b, p, q, x, y) \in [0,1]^6$ satisfying the following collection of inequalities. $b \ge a$ $q \ge p$ $y \ge x$ $a \ge p \ge a^2$ $b \ge q \ge b^2$ $p \ge ...
BPN's user avatar
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3 votes
1 answer
393 views

Weird claims and conclusions in "Introduction to Shape Optimization"

I'm trying to understand the notions of Euler and Hadamard derivatives of shape functionals. All the lecture notes and papers on this topic that I've found seem to build up on the books Shapes and ...
0xbadf00d's user avatar
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4 votes
2 answers
209 views

If all points of a real function with positive values would be local minimum, can one say it is constant function?

During my studies I faced a function $f:\mathbb{R} \to \mathbb{R}^+ $ with the property: for all $x \in \mathbb{R} $ and all $y$ in open interval $(x-\frac{1}{f(x)} ,x+\frac{1}{f(x)}) $ we have $f(x) \...
M. Reza. K's user avatar
2 votes
0 answers
45 views

Notion of distance between linear programs

Consider the linear programming problem \begin{align} \max_{x}&~c^Tx \\~s.t.~~a^Tx &\leq B~,~0\leq x_i \le1 \end{align} where $c$ and $a$ are $n \times 1$ given non-negative vectors. $B$ is a ...
dineshdileep's user avatar
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3 votes
0 answers
68 views

Convergence of iteration of a convex program

Let $\mathbf{V} \in \mathbb{R}_{+}^{n \times m}, \ \ \mathbf{E} \in \mathbb{R}_{+}^{n \times m}$, with $\mathbf{V} \mathbf{1}_{m} = \mathbf{1}_{n}$ and $\mathbf{E}^{T} \mathbf{1}_{n} = \mathbf{1}_{m}$...
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