# Questions tagged [nonlinear-optimization]

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

543
questions

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### Minima of a cdf of multivariate normal distribution with respect to a parameter

Let $\mathrm{X}\sim\mathcal{N}_{3}(\boldsymbol{\mu},\mathrm{\Sigma})$ where
\begin{equation}
\boldsymbol{\mu} = n[(\mu_1-\mu_2)\sqrt{\xi_1\xi_2/(\xi_1+\xi_2)}, (\mu_1-\mu_3)\sqrt{\xi_1\xi_3/(\xi_1+\...

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### Maximize entropy under Kulback-Leibler divergence

I posed this question in math.stackexchange.com, but have not received any answer. I would like to try my luck here.
In this question, it is to solve
\begin{align}
\max_p &-\int dy\,p(y)\ln p(y) \\...

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### Variational problem with constraint

Let $D\subseteq [0,2\pi]\times [0,2\pi]$ and ${D}^\complement$ be the complementary region, i.e.
$D \cup {D}^\complement = [0,2\pi]\times [0,2\pi]$ and
$D\cap {D}^\complement = \emptyset$.
I would ...

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### Relationship between an optimizer and a mode/mean of pdf

Let $X$ be a random variable, uniformly distributed over a support $S$. Let $f(X;\theta)$ be a function of $X$, parameterized by $\theta$. I am hoping to think of a relationship between two quantities:...

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### Matrix relative condition number

I've been working on some distributed optimization problems and faced a bit of a challenge with the following question.
Given $A_1, A_2, .., A_m \in M_n({\mathbb{R})} $ symmetric positive definite ...

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### Efficient way to Non linear constraint programming with polynomials

I have a non-linear programming constraint problem as below:
\begin{equation}
\begin{split}
minimise_{x \in \mathbb{R}^n} &f(x) \\
subject\ to\ &c(x)>= 0\\
&l_i<= x_i <= u_i,\ ...

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154
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### Optimal solution of complex optimization problem

Let $Q(x)=a(x)e^{jb(x)}$ be a complex function of $x$. We want to approximate this function with $R(x)=\alpha e^{jx\beta}$ such that
\begin{align}
\text{arg}\min_{\alpha,\beta} \int_{-\frac{A}{2}}^{\...

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1
answer

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### Round Robin volleyball Tournament [closed]

Consider a set of N teams (N even number) that must make a
Round Robin Tournament. To each pair i; j, i ≠ j, of teams there is associated level
of interest si,j ∈ {1;2;3} of the match between them (1 =...

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### Prove that a strictly monotone function $f : X \to \mathbb{R}$ with convex level curves is a quasiconcave function?

Conjecture: Let $X$ be one of $\mathbb{R^2_0}$ and $\mathbb{R}^2$. Prove that a strictly monotone function $f : X \to \mathbb{R}$ with convex level curves is a quasiconcave function.
Strict ...

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### Can quasi-concave and monotone functions level curves that are not path-connected?

I posted this on MSE [link] and there's been no answer for the past few days. I started a bounty there and decided to post it here as well.
1. For $X = \mathbb{R}^2$, does there exist a quasi-concave ...

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1
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### Point of tangency is an optimal point for a monotone, quasi-concave function

Given $U : \mathbb{R^2} \to \mathbb{R}$ is monotone and quasi-concave, consider the following problem :
$$\max_{(x,y) \ \in \ \mathbb{R}^2}[U(x,y)] \text{ subject to } p_1 x + p_2 y \leq M ; \ (p_1, ...

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### Gap to fill in the Aubin–Ekeland proof of the mountain-pass theorem

Working through the proof of the mountain-pass theorem given in Applied Nonlinear Analysis by Aubin & Ekeland, at what seems to be a critical point of the proof (the top of page 274) they refer to ...

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### Maximization of accumulated entropy of time-changed Brownian motion stopped at $\{-1/2, 1/2\}$

Let $W=(W_t)_{t\ge 0}$ be a standard Brownian motion. Define the running maximum and minimum by $S_t:=\max_{0\le u\le t}W_u$ and $I_t:=\min_{0\le u\le t}W_u$. It is known that
\begin{eqnarray}
&&...

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### Are square configurations the only critical points of the energy on the circle?

$\newcommand{\S}{\mathbb{S}^1}$
$\newcommand{\la}{\lambda}$Let$$M=\{(x_1,x_2,x_3,x_4) \in (\S)^4\,\, |\,\, \text{ all the } x_i \, \text{ are distinct}\} $$
Define $E:M \to \mathbb{R}$ by
$$E(x_1,x_2,...

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### Unique solution to nonlinear optimization through gradient descent

I am trying to estimate the path of a random walk described by the following SSM
$$
\begin{align}
x_{t+1} &= x_{t} + q_{t+1} \newline
y_{t+1} &= h(x_{t+1}) + r_{t+1}
\end{align}
$...

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### Minimize $Q(t,s) = \mathbf{w}^\top \begin{pmatrix} r(t, t) & -r(t, s) \\ -r(t,s) & r(s, s) \end{pmatrix}^{-1} \mathbf{w}$ in $(t, s)$

Let $0 \leq r(t,s) \leq 1$, $t, \ s \in [0, T]$ be a smooth enough function, such that
$r(t,t)$ increases in $t$
$r(t, s) = r(s, t)$ decreases as $t$ and $s$ move away from each other (that is, as $|...

2
votes

0
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### Convergent algorithm for minimizing nonconvex smooth function

Let $\Phi$ be the Gaussian CDF and for $\gamma\ge 0$ and $h>0$, define a loss function $\ell_h:\{\pm 1\} \times \mathbb R$ by
$$
\ell_{\gamma,h}(y,y') := \phi_{\gamma,h}(yy') := \Phi((yy'-\gamma)/h)...

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### Potentially ill-posed nonlinear problem with equality and inequality constraints

I have an optimization problem of the following form:
\begin{align*}
\min_\mathbf{y} &\quad -f(\alpha\mathbf{Ny}|\boldsymbol{\theta})+\alpha\mathbf{p}^\top\mathbf{y} \\
\text{subject to:} &\...

4
votes

2
answers

245
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### The mower's challenge

Weeds have taken over the paths (two squares). If mowed, they don't grow back, but unmowed weeds spread at speed $1$ along the road. What's the minimum speed of the mower to get rid of all the weeds? ...

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2
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154
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### Can we substitute this KKT condition into this optimization problem to reformulate the optimization problem?

Suppose I have the following optimization problem
$$ \min\limits_{\mathbf{x},\mathbf{y}} f(\mathbf{x},\mathbf{y}) \tag{1} $$
It is already known that the target function $f$ is continuous and ...

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0
answers

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### Optimization: Minimization when lambda equals infinity

I would like to estimate function g(x) by the following rules:
$$\hat{g}=\arg \min _{g}\left(\sum_{i=1}^{n}\left(y_{i}-g\left(x_{i}\right)\right)^{2}+\lambda \int\left[g^{(m)}(x)\right]^{2} d x\right)$...

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### Convergence of solutions of regularised least square problems

Let $\mathcal{H}_1$ and $\mathcal{H}_2$ be two Hilbert spaces ($\mathcal{H_2}$ being finite dimensional, but I don't think that it matters). Consider $(A_{\lambda})_{\lambda>0}$ a familly of ...

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1
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139
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### Calculating derivatives of arbitrary-order at an operator's root

Consider roots $f = 0$ of a nicely-behaved real function $f(x, t)$ of two (real) variables.
Namely, points $(x, t)$ on which $f$ vanishes, $f(x, t) = 0$.
Suppose that $x$ can be written as function of ...

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answers

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### Get a specific number of points from a density distribution area to minimize the average distances

Assuming that an area $A$ on the plane has a known density distribution function $\rho (x, y)\geqslant 0$, now the goal is to obtain $n$ points $p_1, p_2, ..., p_n$ on the area so that $\iint_{}^{} \...

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### Minimising risk in dynamical systems

I have been reading the paper of Goerner and Ulancowicz - "Quantifying economic sustainability" in which it is suggested that there is a tradeoff between sustainability and efficiency. ...

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### Optimization over algebraic structures other than matrices?

So I've been spending recently going through optimization literature, and to my understanding, much of statistical learning is just solving the following equation:
$$\min_{\theta} f_{\theta}(x) $$
for ...

3
votes

1
answer

202
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### Obtaining the "best possible" inequality by tuning hyper-parameters

I encountered the following problem in one of my research projects which can be encapsulated as follows. Let's say we have a set $\mathcal{C}$ of functions $f$ defined from $\mathbb R_+$ to $\mathbb R$...

2
votes

2
answers

133
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### Analytic solution of low-dimensional Riccati equation

Consider the nonlinear map $F_i:\mathbb R^2 \to \mathbb R$
$F_i(x):=\varepsilon^2\langle x, A_i x\rangle +\varepsilon\langle b_i,x \rangle + x_i,$
where $A_i$ is some matrix and $b_i$ some vector
Can ...

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0
answers

82
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### Solution of a simple optimization problem

Let $\mathbf{U}_1$ and $\mathbf{U}_2$ be two arbitrary unitary matrices and $\mathbf{D}$ be a diagonal matrix. What is the solution of the following optimization problem?
\begin{align}
\min_{\mathbf{...

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votes

1
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44
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### Formulating a problem as a mixed-integer conic program

I have the following integer optimisation problem, and I wonder whether it can be reformulated as a conic program that can be solved with, e.g., Mosek. Suppose the $n$-dimensional vectors $a, b$ and $...

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0
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### The best unitary matrices that approximate a matrix product

Let $\mathbf{A}$ be an arbitrary $N\times N$ complex matrix. Moreover, $\mathcal{U}_1$ and $\mathcal{U}_2$ are distinct subsets of all unitary matrices. Suppose the matrices $\mathbf{U}_1$ and $\...

4
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1
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239
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### Which set of functions admits the existence of the minimizer?

Let $a,b \in \mathbb R$ and consider the functional $J$ on $X$:
$$J[u] = \int_0^1 \left( (u'(x))^2 -a)^2 + b \ln (1+ u^2(x))\right) dx$$
Providing reasons specify if the $\inf J$ over $X$ is attained ...

1
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2
answers

238
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### Optimization of a integral function

I have a function $h(y,x_1,x_2,\ldots,x_n)$. It is known that the minimum value of $h$ for any $y$ is attained when $x_1 = x_n$ and $x_2 = x_3 = \cdots = x_{n-1}$. Now consider the following function
\...

2
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1
answer

102
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### The regularity theorem, a non-regular minimizer problem

During my self study to the calculus of variations I come across this problem. Because of my search, I know what I wanted to do but I need some help to do them.
The function $f:[-1,1] \times \mathbb R ...

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0
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103
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### Convergence of ODE solutions almost everywhere to a stable equilibrium point

Theorem: Suppose ${\bf g} :\mathbb{R}^n \mapsto \mathbb{R}^n$ is continuously differentiable, there exists a set $\mathcal{A} \subset \mathbb{R}^n$ such that $\bf g$ is uniformly Lipschitz on $\...

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2
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### Integrals can sometimes be computed through their saddle points. Are there examples of converse, when saddle points are found via integrals?

Under some reasonable assumptions integrals with large exponents can often be computed via saddle point approximations, e.g.
$$\int e^{-\lambda f(x)}\approx e^{-\lambda f(x_0)},\qquad \lambda\to\infty$...

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0
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### Closed form expression for an opt-problem

Consider the following optimization problem
\begin{align}
G(t) = &\max_{x\in R^N} ~~x^\top P x\\
&\mbox{subject to}\\
&\hspace{1cm} x^\top P x \leq t\\
&\hspace{1cm} x^\top x \leq 1,
\...

3
votes

1
answer

229
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### Special version of Tonelli’s theorem

I am trying to prove this theorem. I have not found anything similar to it on the internet.
Special version of Tonelli’s theorem
Assume that the functions $f(x,u): [a,b] \times \mathbb{R} \to \mathbb{...

1
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1
answer

176
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### Proof of extended version of non-random "almost supermartingale"

In this question, a non-random version of "almost supermartingale" theorem is proved.
Here, I would like to extend/apply the non-random version to the slightly different situation. I wonder ...

2
votes

1
answer

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### Why is the Ekeland variational principle called a principle? [closed]

Why is the Ekeland variational theorem called the Ekeland variational principle?
I think (or maybe I studied somewhere) this is because of its equivalency with the Takahashi theorem, the Caristi ...

1
vote

1
answer

170
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### Can we invoke "almost supermartingale" Theorem for deterministic sequences?

Perhaps stupid question.
Question: Can "almost supermartingale" theorem be equally applicable to prove the convergence of some algorithms solving non-random optimization problems?
Attempt ...

2
votes

1
answer

113
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### On the Lipschitz continuity of $x \mapsto \arg\min_{c \in C}d(x,c)$ w.r.t Hausdorff distance

Let $C$ be a (nonempty) compact subset of euclidean $\mathbb R^n$, and consider the set-valued map $p_C:\mathbb R^n \to 2^C$ defined by
$$
p_C(x) = \{c \in C \mid \|x-c\| = \mbox{dist}(x,C)\},
$$
...

3
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0
answers

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### What is the name for this type of optimization problem?

As we all know, a classic optimization problem can be represented in the following way:
Given a function $f: A \to \mathbb{R}$, find an element $x_0 \in A$ such that $f(x_0) \le f(x)$ for all $x \in ...

3
votes

5
answers

392
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### Reference request: importance of Lipschitz continuity

I see that Lipschitz continuity is a common assumption used in optimisation, statistics, machine learning, etc.
Could you point me in the direction of some literature that discusses why Lipschitz ...

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0
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46
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### How to find a set given its support function

Let $\mathcal{U}$ be a convex and compact set. Its support function is defined as $\delta^*(v|\mathcal{U})=\sup_{u\in \mathcal{U}} v^T u$. Assume that we are given the support function $\delta^*(v|\...

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votes

1
answer

20
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### 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 ...

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0
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### 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 ...

2
votes

1
answer

61
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### 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
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### 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 ∈ ...

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0
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### 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 ...