# Questions tagged [nonlinear-optimization]

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

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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) \\...
1 vote
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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:...
1 vote
127 views

### 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,\ ...
154 views

### 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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### 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 ...
1 vote
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1 vote
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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:} &\...
245 views

### 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? ...
1 vote
154 views

### 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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1 vote
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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 ...
88 views

### 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
170 views

### 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 ...
113 views

### 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 $$p_C(x) = \{c \in C \mid \|x-c\| = \mbox{dist}(x,C)\},$$ ...
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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 ...
31 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  in different ways. However, since I cannot find a pdf version of this paper and ...
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 (...