# Questions tagged [nonlinear-optimization]

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

491
questions

**-1**

votes

**0**answers

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 ...

**2**

votes

**0**answers

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-...

**3**

votes

**1**answer

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) ...

**1**

vote

**1**answer

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_{...

**0**

votes

**1**answer

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 ...

**2**

votes

**1**answer

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 ...

**1**

vote

**1**answer

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=...

**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

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 ...

**1**

vote

**0**answers

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)...

**4**

votes

**0**answers

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 ...

**3**

votes

**2**answers

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)...

**0**

votes

**0**answers

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 ...

**2**

votes

**0**answers

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\...

**1**

vote

**1**answer

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 ...

**0**

votes

**1**answer

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-...

**1**

vote

**0**answers

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 ...

**-1**

votes

**1**answer

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, ...

**0**

votes

**0**answers

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\}...

**2**

votes

**1**answer

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 ...

**3**

votes

**1**answer

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(...

**0**

votes

**0**answers

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**

votes

**1**answer

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)/...

**2**

votes

**0**answers

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=...

**5**

votes

**1**answer

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=...

**1**

vote

**2**answers

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 ...

**2**

votes

**0**answers

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},\...

**0**

votes

**0**answers

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)...

**2**

votes

**0**answers

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 ...

**0**

votes

**0**answers

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 $y(\theta)$ ($\...

**1**

vote

**0**answers

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 \...

**0**

votes

**0**answers

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\...

**1**

vote

**0**answers

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,...

**0**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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 $...

**1**

vote

**0**answers

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 ...

**2**

votes

**0**answers

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 ...

**0**

votes

**0**answers

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 ...

**1**

vote

**1**answer

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,+\...

**5**

votes

**5**answers

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 ...

**0**

votes

**0**answers

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\...

**7**

votes

**1**answer

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)...

**0**

votes

**0**answers

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**

votes

**1**answer

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 ...

**0**

votes

**0**answers

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 $r$ ...

**7**

votes

**2**answers

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 $x_1,\dots,x_n \ge 0$?
...

**1**

vote

**1**answer

123 views

### Dual problem with integrals

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

**0**

votes

**0**answers

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**

votes

**0**answers

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+\...

**0**

votes

**1**answer

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 \...