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

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Recovering a partition from spectral properties of the graph Laplacian

Let $G$ be a weighted graph with vertices $V$. Let $W$ be its real-valued, non-negative, $|V|\times|V|$ adjacency/affinity matrix. Let $L = \mathrm{diag}(W\mathbf1)-W$ be the (unnormalized) graph ...
3
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1answer
198 views

Optimization problem - maximizing number of satisfied linear inequalities subject to a quadratic constraint

I am wondering what is known about optimization problems of the following type. Our control x is a unit vector in $\mathbb{R}^n$. We are given a finite number of linear inequalities $$Az≥b,$$ and we ...
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66 views

Big eigenvalues of a special stochastic matrix

Given a matrix $M$ of size $n\times n,$ we write its different eigenvalues by $x_1,x_2,\ldots,x_m$ with $m\leq n$ such that $|x_1|>|x_2|>|x_3|>\cdots|x_m|,$ and call $x_2\doteq ...
3
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1answer
72 views

Conjugate gradient algorithm where first search direction is not equal to residual

In usual formulation of conjugate gradient algorithm initial search direction is taken to be the residual (so residual and search direction spans Krylov subspace). However, in cases where inexact ...
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140 views

Quadratic optimization with parameter in constraint

Disclaimer: I posted the same question on math.stackexchange. However, the FAQ suggests to post research-level questions in this forum. Question: Given a function $q: \mathbb R^{N\times N}\mapsto ...
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3answers
149 views

Constraint optimization problem for any dimensionality $n>1$.

I am going to post a particular example for the sake of clarity. One needs to maximize a real function $F = a_1a_2 + a_2a_3 + \cdots + a_{n - > 1}a_n + a_na_1;$ with active ...
3
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1answer
448 views

The average number of people that can sit on a bench of a given length.

Let me explain what I mean: The width of the average person varies, perhaps with a normal distribution. Given a specific variance, how many people (on average) can sit side-by-side on a bench of a ...
2
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1answer
357 views

Maximizing supermodular functions

I have a real supermodular objective function which I want to maximize with constraint. The constraint is on the size, like |A|=k . I am wondering if anyone can give me more information about a ...
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194 views

minimization of a function when the feasible set is an unbounded cone

I have the following semi-infinite programming problem: I need to minimize a strictly convex real-valued function $f:\mathbb R^n\to\mathbb R$ subject to infinite linear constraints. I know in advance ...
2
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1answer
272 views

lipschitz constant of a multivariate function

I have a function $f:\mathbb{R}^{50} \rightarrow \mathbb{R}$ and I need to compute the Lipschitz constant of $f$ to solve an optimization problem using a specific algorithm. Does any one have ...
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54 views

functional maximization

Define a functional space of functions of the form $F(t)=p_1 exp^{-\mu_1(\delta-t)}+p'_1 (1-exp^{-\mu_1(\delta-t)}))$. $p_1,p'_1,\delta,\mu$ are parameters in [0,1] and trivially, variation of these ...
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240 views

Generalization of the equilateral triangle?

I consider points in the two-dimensional plane. An equilateral triangle is a set of three points in the plane which are equidistant. Suppose now I have $n$ points $x_1,...,x_n$. What is the ...
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1answer
130 views

What kind is this optimization problem

I come across a problem like $\max {\frac{1+v}{1-u}}$ $s.t.~$ $ux^2+vy^2-xy\ge0$ $\forall x,y\in\mathbb{R}$ I do not know much of optimization. What I have done is that $ux^2+vy^2\ge ...
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0answers
54 views

Minimizing inside a spherical uncertainty region

I am trying to figure out how to solve: $\min_U r_{p}$ where $r_{p}=\alpha^\intercal\omega$ and $U$ is a sphere centered at $\alpha$ with radius equal to $\chi|\alpha|$ . ( $\omega$ is a vector or ...
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2answers
172 views

Convex optimization problem to QPP

Briefly, have the following problem: \begin{equation} \sum_{i = 0}^n a_i \ (max [ F_i( \bar x ), 0 ] )^2 \rightarrow min, \\\\ s.t.\\\\ A \bar x \leq b \end{equation} where $ F( \bar x ) $ is a ...
5
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1answer
307 views

Solve equation with matrix variable

I want to solve a matrix $\Omega$ from a equation $\sum_k (\Omega + \Theta_k)^{-1} = Q$. The $Q$ and $\Theta, \forall k=1...K$ are known, and are positive definite matrices. $\Omega$ also has to be ...
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1answer
167 views

A certain type of quadratic problem.

I am interested in solving the following equality constrained quadratic (?) problem. \begin{align} \min_{u^{H}u=1}~(u^{H}A_1u) \\\ s.t.~ u^{H}A_2u=0 \end{align} $A_1$ and $A_2$ are $N\times N$ ...
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2answers
525 views

Quadratic Farkas' Lemma?

The Farkas Lemma says that if a system of linear inequalities implies yet another linear inequality, then this last inequality can be obtained by taking a positive linear combination of the ...
5
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2answers
262 views

Simultaneous maximization of two Generalized Rayleigh Ritz Ratios

Consider hermitian positive semi-definite matrices $A_1$ and $A_2$. Consider also positive definite matrices $B_1$ and $B_2$. I want to maximize the minimum of the two Generalized Rayleigh Ritz ratios ...
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1answer
127 views

Nonlinear matrix equation 2

Solve the following nonlinear equations for $v$ and $w$ $Avv^TAw+Bvv^TBw=\lambda_1v+\lambda_2w$ $Aww^TAv+Bww^TBv=\lambda_1w+\lambda_2v$ $v^Tw=w^Tv=0$ $v^Tv=w^Tw=1$ where $\lambda_1, \lambda_2, ...
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1answer
205 views

Does Quadratic Programming get easier when it's described by a diagonal matrix?

Generally, Quadratic Programming solves the problem $$\text{Given }Q, c, A, b,\text{ choose }x \text{ to maximize } x^TQx + c^Tx \text{ subject to } Ax \le b$$ In this form, Quadratic Programming is ...
2
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2answers
164 views

product of variables in objective function

Hi there, I'm looking for a solver that allows me to solve an optimization problem of the form min x1*x2*x3,...,xn subject to some linear constraints. I've used gurobi before, however I couldn't find ...