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

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6
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0answers
65 views

Bounding volume of cell in complement of zero set

I am given an integer polynomial $f \in \mathbb{Z}[X_1, \ldots, X_n]$ of bounded degree and bounded coefficient size. The polynomial's zero set partitions $\mathbb{R}^n$ into cells. What I am looking ...
5
votes
0answers
178 views

Characterizing matrices with rank constraint

Given matrix $M\in\Bbb\{0,1\}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M]=\{Q\in\Bbb\{0,1\}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad M-Q\in\Bbb\{0,1\}^{n\times ...
5
votes
0answers
248 views

Find subset of collection of sets whose intersection has minimum average value

Let $a_1,\ldots,a_n>0$, and let $S_1,\ldots,S_d\subset\{1,\ldots,n\}$ (all non-empty). For any $I\subseteq\{1,\ldots,d\}$, define $S(I)=\bigcap_{i\in I} S_i$. Given some $1\leq s < d$, consider ...
4
votes
0answers
169 views

Comparison of Constrained Optimization Methods

I am trying to solve a constrained optimization problem using filter methods and came across two papers on the topic that I am having some problems with. The original filter method paper is the ...
4
votes
0answers
70 views

max-min optimization problem

I'm curious if there is any nice way to approach solving the following kind of optimization problem. Given a $n \times m$ matrix $A = (a_{ij})$, I want to solve \begin{align*} & \max_{c}\min_{1 ...
4
votes
0answers
139 views

maximize non-convex composite function

I want to maximize a composite function over a convex set \begin{equation} \begin{aligned} & \underset{\mathbf{p}}{\text{maximize}} & & f(\mathbf{p})-g(\mathbf{p})\\ & \text{subject ...
4
votes
0answers
41 views

How does one go from convexity to submodularity?

If I have a function which is convex in the hypercube, $[-1,1]^n$ then when would it imply that its restriction to $\{-1,1\}^n$ is submodular? It would be helpful is someone can share some specific ...
4
votes
0answers
156 views

Optimization problem involving Multivariate Normal

I use $\phi(t)$ to describe the standard normal distribution density and $\Phi(t)$ as the normal distribution CDF and would like to prove that for all $n\geq3$, the function: ...
3
votes
0answers
94 views

Determining Nullspace Basis such that only one column is deleted or added as row is added or deleted, and remaining columns of basis stay the same

I would like to compute, in MATLAB, the basis Z for the nullspace of an m by n matrix A, such that if one row of A is added (resulting in A_a), the basis for A_a is n-m-1 of the n-m columns of Z, ...
3
votes
0answers
129 views

existence of optimal control

I'm looking for an existential result in optimal control for the following class of problems: Given $T > 0$, $\bar x, \hat x\in\mathbb R^d$, an instantaneous cost function $c:\mathbb R^d\times ...
3
votes
0answers
376 views

An intuition for three different types of subgradients (proximal, regular, limiting)

I'm having a bit of difficulty getting my head around the different types of subgradients we're currently covering in a nonsmooth optimisation class I'm taking. These subgradients are (assume $x \in$ ...
2
votes
0answers
45 views

How to approximate higher-degree multivariate polynomial in space of lower-degree multivariate polynomials with some constraints?

For a polynomial $P_{1}(x)$, $x\in {\mathbb R}^n$ with a higher-degree, how to find a lower-degree polynomial $P_{2}(x)$ with determined structure or bounded degree to approximate it with the ...
2
votes
0answers
64 views

Learning rule for recurrent neural network with flexible time steps

Summary: I want to train a recurrent network to output some answers, but the recurrent network is allowed to re-iterate through itself a flexible number of times for each input-output pair. Why this ...
2
votes
0answers
80 views

Minimize L-infinity norm with restrictions

I need to minimize the following L-infinity norm with respective to $\tau$. L-infinity norm of a matrix $A$ is defined as $\|A\| = max_{i,j}|a_{i,j}|$. $$ min_{\tau} \| I -S(S+\tau)^{-1}\| $$ $$ ...
2
votes
0answers
50 views

Maximizing a convex bounded function of a PSD matrix

Let $f(X)$ be convex and continuous function , with $X$ a PSD matrix. Assume that under the affine set of constraints $\mathcal{A}(X)=b$ and the convex constraint $f(X)\le1$ there is an optimal, ...
2
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0answers
134 views

Why hexagons? The maximal minimum of a sum of cosines on the plane with frequencies on the unit circle

We wish to maximize the minimum of a weighted sum of cosines in the plane, when the frequency components are on the unit circle. Formally: $$\max_{\{ a_i,\theta_i,\phi_i \}_{i=1}^{N} } \min_{(x,y) ...
2
votes
0answers
65 views

Sequence transformations that are entropy invariant

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Define entropy of $\mathcal{A}=\{a_i\}_{i=1}^m$ by ...
2
votes
0answers
99 views

Containing a “fuzzy” ellipsoid within an ordinary ellipsoid

Consider the ellipsoid described by the inequality $(x - x_c)^T P^{-1} (x - x_c) \leq 1$, where the vector $x_c \in \mathbb{R}^n$ denotes the center of the ellipsoid and the symmetric positive ...
2
votes
0answers
84 views

Techniques for proving that a set of constraints over the integers are inconsistent

I have a problem which boils down to showing that a set of constraints has no solutions. A simplified version of this constraint system would be the following system: $$ \left\{ \begin{array}{l} ...
2
votes
0answers
166 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 ...
2
votes
0answers
235 views

Find minimum-area ellipse enclosing a set of ellipses, all centered at the origin

Given a set of N > 2 (two-dimensional and coplanar) ellipses, all centered at the origin, how do I find the ellipse with the minimum area which encloses all of them? Background: Thanks to Will Jagy ...
1
vote
0answers
12 views

Deterministic global solution to find the Optimal-knot placements for continuous piecewise linear functions to fit nonlinear data

I have been searching lately for a deterministic global technique to linearize a nonlinear function with continuous piecewise linear regions. I've a univariate nonlinear function y=f(x). where f(x) ...
1
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0answers
79 views

Minimization of nonlinear integral operator

For non-negative self-adjoint traceclass operators $0\leq T \leq 1$ with $\mathrm{tr}T=N$ on the Hilbert space $L^2(\mathbb{R}^3)$ s.t. $\iint |\nabla T^\alpha(x,y)|^2 dx dy <\infty$, I would like ...
1
vote
0answers
99 views

Lower bound on the value $\textbf{1}^Tx$ such as $Ax\geq b$

The problem may be formulated as follows: We are given a set of $m$ positive numbers $\{b_1,...,b_m\}$ and a set of $n$ positive numbers $\{v_1,...,v_n\}$. We have $v_j\leq K$, $j=1,...,n$, for a ...
1
vote
0answers
71 views

reconstructing a linear order corrupted by noise

Suppose we have a partial order (efficiently computable), $\leq$, on $\mathbb{R}^n$, a set $S \subset \mathbb{R}^n$, and let $\rho$ be the standard Euclidean metric. We want to find a set $S^\prime = ...
1
vote
0answers
71 views

A (non-convex) minimization quadratic programming problem with d constraints

Minimize $0<\omega_{dd}<2$ subject to $$\sum_{j=1}^{d}(\omega_{dj} - \omega_{ij})^{2} \geq 4, i=0,1,...,d-1,$$ where $-2<\omega_{ij}<2$ is known for $0 \leq i \leq d-1$ and $1 \leq j \leq ...
1
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0answers
63 views

Optimization question: maximize quadratic objective with semidefinite constraints

I recently encountered the following optimization problem: $\max \|AX\|_F^2$ subject to: $X\succeq0$ and $Xb_i\leq c_i$ for a collection of $T$ conditions: $i=1,\ldots,T$. Matrices $A$ and $X$ are ...
1
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0answers
40 views

Frobenius nearest non-negative Gram matrix of balanced row-sums

Let $W \in \mathbb{R}^{n \times n}$ be any non-negative real symmetric matrix. For $k \leq n$, let $\mathcal{F} := \{X \in \mathbb{R}^{n \times k} \ | \ X \geq 0, X \mathbf{1} = \alpha \mathbf{1}, ...
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0answers
31 views

Minimizing sum of functions, while keeping their values non-negative

Suppose we have data $\mathbf{x}_i$, $i\in \{1, \ldots, K\}$, and we're trying to find parameters $\hat{\mathbf{\theta}}$ such that $$\hat{\mathbf{\theta}} = \underset{\mathbf{\theta}} ...
1
vote
0answers
86 views

Likelihood convexification

I am doing constrained vector optimization using a non-convex non-linear likelihood function. My problem is of the following form: $$\begin{align*}\hat Q &= \underset{\vec Q}{\arg\min} -\log ...
1
vote
0answers
70 views

modern exposition of exact ground state of classical XY model or Ising model

What is the state of art technique in solving exact ground state of Heisenberg model, meaning minimization of the H terms (hamiltonian) provided infinite spin space? ...
1
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0answers
56 views

Optimization of a multilinear function over a product of hypersimplices

Let $P = \Delta_1 \times \cdots \times \Delta_N$ be the Cartesian product of $N$ hypersimplices. Let $f : P \to \mathbb{R}$ be a multilinear function of $N$ variables, ie $x_i \mapsto f(x_1, \ldots, ...
1
vote
0answers
42 views

Limiting Entropy of deterministic sequences - 2

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_a$ be limiting distribution ...
1
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0answers
42 views

Changing a nonlinear equality constraint into some conic inequality plus rank constraint

If we have a constraint optimization problem in which one of our constraint is $\prod\limits_{k = 1}^N {\left( {x - {a_k}} \right) = 0} $ . How could this nonlinear equality condition be changed into ...
1
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0answers
68 views

Estimation of part of parameters from an ODE

Suppose, we have an ODE $$ \frac{dy}{dt}= f(t,y;p',a)$$ or alternatively $$ \frac{dy}{dt}= f(t,y;p)$$ where the set of all parameters $p = (p',a)$. We only need to estimate part of parameter set ...
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0answers
401 views

Subtour Elimination in Travelling Salesman Problem using MTZ

I am trying to formulation a problem similar to a Traveling Salesman with Time Window constraints. To eliminate subtours, I need to use some constraint similar to a generalization of MTZ constraints ...
1
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0answers
157 views

Constructing an $\epsilon$-net for a Lipschitz subspace of $L^2$

Let $X$ be a subset of $L^2([0,1])$ which contains only Lipschitz function. Also, the member of $X$ are uniformly bounded $$ |x(t)| < M, \text{ for all $x \in X$ and $t \in [0,1]$}. $$ Let $F: X ...
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0answers
140 views

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 ...
1
vote
0answers
534 views

Solving a system of complex non-linear equations

I have a set of five equations which can be described as follows: $m_{i}=\frac{k_{1}}{(x+a)^{i}} + \frac{k_{2}}{(b+d)^{i}}+ \frac{k_{3}}{c^{i}}$ for i=1 to 5 where $$\eqalign{ ...
1
vote
0answers
229 views

Multiobjective semidefinite programming

Let $C$ be size $n \times n^{2}$. Let $B$ be size $2^{g(n)} \times n^{2}$ where $g(n) > n$. There is only one $\mathcal{1}$ per row of $C$ and remaining entries of $C$ are $\mathcal{0}$. $B$ is ...
0
votes
0answers
17 views

Generalized Lax-Milgram for Weak Formulation of 1D Linear Schrodinger

I am interested in the variational formulation of the 1D Schrodinger equation: $i u_t- \beta u_{xx} = 0 $ and $u(x,0)=u_0(x)$ which upon integration by parts yields: $i(u_t,v) + \beta (u_x,v_x) = 0$ ...
0
votes
0answers
25 views

Nuclear norm maximization

I am trying to solve a nuclear norm maximization problem: $$\arg \max_{Q \in O(n)} \|WQV^T\|_*$$ where $Q$ is an $n \times n$ orthogonal matrix and $W$ and $V$ are real $d \times n$ matrices. I've ...
0
votes
0answers
20 views

Correct definition of submodularity

I am currently looking at a paper whose submodularity definition is different from whatever I thought I knew. In this paper, the authors consider a function $\Pi_2(q;a^r)$, where $q$ is composed of ...
0
votes
0answers
30 views

Question on solving an optimization problem using Variable splitting and ADMM

Tell me if I have found the right approach to the following optimization problem: $$ min_{x} \frac{1}{2}\left \| Ax-b \right \|_2^2 \\ s.t. \ \ \Phi v=x \ , \ {x^T(1-x)}=0 $$ $A$ and $\Phi$ ...
0
votes
0answers
9 views

Binary Nonlinear Optimization Problem Transform to Continuous Form

Hi I have a mixed binary NLP problem \begin{equation} min\: f(x) \end{equation} s.t. $$ g_i(x)<=0 $$ $$ h_j(x)=0 $$ $$ x_k=1\text{ or }0 $$ I know the optimal of f(x) is approximately 0, so can I ...
0
votes
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27 views

Is this QCQP convex or nonconvex

\begin{equation} \begin{split} \min_{x\in \mathbb{R}^n}\:f(x)=(1/2)x^{T}Q_0x+c_0^T x \end{split} \end{equation} s.t. $$ g_i(x)=\frac{1}{2}x^T Q_ix-lmax_i\leq0,i\in\{1,...,m/2\} $$ $$ ...
0
votes
0answers
23 views

solution of an infinite horizon optimization problem

Give the following formulation: $\min_{\{x_s(t):\forall s,t\}} \sum_{s \in \mathcal{S}} \mathbf{1}\left(\lim_{T\rightarrow \infty} \frac{1}{T} \sum_{t=1}^T \frac{y_s(t)}{x_s(t)}\leq 1\right)$ $s.t. ...
0
votes
0answers
21 views

Maximizing modular function subject to supermodular constaint

I'm trying to solve a constrained optimization problem with submodular functions and get some nice properties of the solution. Unfortunately, I think I am in a setting where Topkis' theorem does not ...
0
votes
0answers
43 views

How to use the property of Frobenius norm in this proof?

Let $A \in \mathcal{S}^{n}$($\mathcal{S}^{n}$ is the $n \times n$ symmetric matrix space), $P,Q \in\mathbb R^{n \times n}$, and $\rho \geq 0$ be given. Show that: $$A \succeq P^{T}ZQ+Q^{T}Z^{T}P$$ for ...
0
votes
0answers
142 views

Asymptotics to Taylor expansions?

I posted a question on MSE about approximating Taylor series but Despite a bounty I did not receive any answers or comments. Maybe you guys can help. ...