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

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0
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29 views

how to solve a linear equation (Ax-b)T- lamda(c)?

I'm trying to solve an linear optimization problem, it's first order Lagrange condition leas to this equation $$ (Ax-b)^T A- \lambda C = 0. $$ Here $A$ is an $m\times n$ matrix, $m>n$, $C$ ...
0
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0answers
19 views

Interior point V.S sqp algorithm for large scale optimization

I have an large scale optimization problem that works with fmincon solver within sqp algorithm. But it is so slow. However, ...
2
votes
1answer
58 views

Rate of convergence for cyclic gradient descent

I'm trying to solve the optimization problem $\min_x \frac{1}{n} \sum_{i=1}^n f_i(x)$ where $f_i$ are (strongly) convex, smooth, lower semi-continuous, etc. However, I am not able to do conventional ...
2
votes
0answers
225 views

Programming workbooks in C++ and Research Math [closed]

I know the basics of C++ by taking a few courses and going through "C++ Primer" by Lippman. As a math graduate student, I would love to get my hands on some programming-math exercises geared towards ...
2
votes
1answer
46 views

When is a convex program continuous in its constraint vectors?

Consider $$F(z)=\min ae^{-x}+b e^{-y} s.t. x\ge 0, y\ge 0\text{ and } x+y=z$$ I checked if this function is continuous, but it is not at $z=0$. $F(z)=2\sqrt{ab}e^{-z/2}$ when $z\ne 0$, and ...
0
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0answers
78 views

Global and local maxima in a weighted sum of logarithms of linear functionals?

Constrained Optimization Problem Is is possible to describe, and locate efficiently, the maxima of the function $f$, as described below in the parameters $\mathbf{x} = (x_1,...,x_m)$. The constraints ...
1
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0answers
28 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}} ...
0
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0answers
58 views

Can this equation have an explicit solution?

Given $n > 0$, $0 \leq i \leq n$ is an integer, $D = diag(d_1, \dots, d_n)$ is positive definite, $e_i$ is the $i$th column of a $n \times n$ identity matrix, $u \in R^n$ such that $B = D + u * ...
3
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0answers
105 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 ...
0
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0answers
17 views

Is it possible to design an algorithm an optimization problem with non-convex and coupled constraints, using decomposition methods?

Recently, I am working on an optimization problem in wireless networks. I formulate the problem to be an optimization problem with non-convex and coupled non-linear constraints. And I want to design a ...
3
votes
1answer
152 views

Find a line such that sum of perpendicular distances of points to the line is minimized

Given a set of points (column vectors) $S = \{p_1, p_2, \cdots, p_n\} \subset \Re^d$, let $A \in \Re^{n \times d}$ be a matrix of which each row is just $p_i^T$. It is easy to find a unit vector $s_1$ ...
3
votes
1answer
60 views

Constrained vs Unconstrained Optimization

I'm currently working on an optimization problem with a linear objective with linear and nonlinear constraints, i'm facing difficulties reaching a good solution, so i was advised to move the nonlinear ...
1
vote
0answers
72 views

Minimize Product of Sums of Squared Distances

The Question Given two sets of vectors $S_1$ and $S_2$,we want to find a unit vector $s$ such that $$\{\sum_{u\in S_1}(\|u\|^2-\langle u, s \rangle^2)\} \cdot \{\sum_{v\in S_2}(\|v\|^2 - \langle v, ...
0
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0answers
20 views

Complexity of optimizing a bi-objective function with integer constraints

I have two different objective functions, each of which can be solved optimally in polynomial time. Does this mean, I can optimize a linear combination of these objective functions in polynomial time ...
0
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0answers
100 views

How to decide a value of learning rate for Stochastic Gradient Descent?

I'd like to know how to decide a value of learning rate for Stochastic Gradient Descent (SGD), such as $\eta$ on the following parameter update iteration equation, $w_{i+1} = w_i + -\eta \nabla ...
4
votes
2answers
107 views

Convexity of a (non-symmetric) function of matrices

Let $f : H_{n\times n} (\mathbb{C}) \rightarrow \mathbb{R}$ be the function on Hermitian, positive semidefinite matrices $f(A) = \frac{M_i (A)}{\det(A)}$ where $M_i(A)$ is the determinant of the the ...
3
votes
1answer
65 views

Analysis of first-order methods for constrained convex optimization with approximate oracles

In many first-order optimization methods an oracle is needed whose action enforces the constraint/regularizations. For example, in projected gradient descent, conditional gradient method, and proximal ...
2
votes
1answer
107 views

Optimization over symmetric polynomials

Consider the following constraint satisfaction problem: Let $\alpha_1 , \ldots, \alpha_k \in \mathbb{R}$ be given as well as an error parameter $\epsilon$. Find $p_1, \ldots, p_n$ such that (i) $0 ...
5
votes
0answers
163 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 ...
6
votes
1answer
465 views

Bound on the sum of arguments

Problem: Show that for all real $s,t,u$ and all complex $z$ with $|z|<1$ one has $$(*)\qquad \arg\frac{1-zf(s-u)}{1-zf(s+u)} +\arg\frac{1-zf(t+u)}{1-zf(t-u)}<\pi, $$ where $f$ is the ...
1
vote
0answers
37 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 ...
2
votes
0answers
64 views

How to solve the following generalized quadratic programming problem [closed]

I want to solve a generalized form of a quadratic programming problem $$\min_x \left(\sqrt{x^TPx}+\sqrt{x^TQx}\right)^2+c^Tx$$, $$\textrm{ s.t. } Ax\le b.$$ Here, $P$ and $Q$ are both positive ...
0
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0answers
63 views

Specific optimization problem solution procedures

Is there a standard procedure to solve following two optimization problems? $$\mathsf{Problem\mbox{ }I}:\mbox{ }\min_{A\in\{0,1\}^{n\times n}:rk(A)=r}\mbox{ }\max_{R,S\in\Bbb R^{n\times ...
4
votes
2answers
217 views

Convexity of a function of matrices

Let $A$ be an $n\times n$ positive-definite matrix. Let $0<\lambda _1 \leq \lambda_2 \leq \lambda _3 \ldots \leq \lambda _n$ be the eigenvalues of $A$. Let $n\geq k\geq 1$. Is the function $f(A) = ...
1
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0answers
48 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? ...
5
votes
0answers
185 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 ...
1
vote
0answers
32 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, ...
0
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0answers
28 views

non-coherent estimation problem

I have the following signals $$\left[\begin{array}{c} y_{mn} \\ y_{nm}\end{array}\right] =\left[\begin{array}{c} x_{n} \\ x_{m}\end{array}\right]h_{nm} +\left[\begin{array}{c} e_{mn} \\ ...
2
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0answers
118 views

Solution of a linearly constrained quadratic programming problem [closed]

What is the solution of the following optimization problem: \begin{align} &\min{\mathbf{p}^\mathrm{T} \mathbf{B} \mathbf{p}}\\ &\text{subject to}: \mathbf{0}\leq{\mathbf{p}}\leq \mathbf{1}. ...
2
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0answers
47 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 ...
5
votes
1answer
165 views

Finding sparsest solution of a linear system

I want to find the solution with most zero-components for the following problem: $Ax=b$ for $A\in \mathbb{R}^{k\times n}, b \in \mathbb{R}^{k},k<n$, where $x$ is real and has no additional ...
1
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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
vote
1answer
105 views

Limiting Entropy of deterministic sequences - 1

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$. Consider distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_{a,m}$ be distribution at ...
1
vote
1answer
78 views

Entropy difference dominance of sequences

Consider a collection of positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Similarly for the collection $\{a_i\}_{i=1}^{m+1}$ form the distribution ...
1
vote
1answer
56 views

Entropy dominance of certain restricted sequenes

Say you have positive $\{a_i\}_{i=1}^n$ and you have $p_i=\frac{a_i}{\sum_{i=1}^na_i}$, then assume you have a $C$ such that $C<2a_n\ll\sum_{i=1}^na_i$ (that is $C$ is not very large), then define ...
2
votes
1answer
68 views

Entropy dominance

Let $0<a<b<c$ be distinct positive reals. Define four different probability distributions: $$\mathcal{P}_{ab}:P_{a,ab}=\frac{a}{a+b}=1-P_{b,ab}$$ ...
1
vote
1answer
57 views

Maximizing a certain concave function over a non-convex set

I have been working on a problem which involves maximizing a concave function over a non convex constraint set. The problem is the following $$max. \frac{1}{2}x^TBx\\ s.t.\ x_i(1-x_i)=0,\ \ ...
1
vote
1answer
74 views

Smooth unit vector field on a tetrahedron to interpolate vertex constraints

For a tetrahedron $T\subset \mathbb{R}^3$ with vertices $r_i\in \mathbb{R}^3$ , $i=1,\ldots,4$, and unit vectors $u_i\in \mathbb{S}^2$ at each vertex $i=1,\ldots,4$ consider the (energy) functional ...
11
votes
2answers
577 views

Conjecture on maximum of symmetric combinatoric function

A curious symmetric function crossed my way in some quantum mechanics calculations, and I'm interested its maximum value (for which I do have a conjecture). (The question was first asked at math.SE, ...
0
votes
1answer
59 views

Finding maximum of a function with unfixed number of variables

Can anybody solve this: For a constant positive integer $n\geq6$ find $k$ and positive integers $a_{1},a_{2},...,a_{k}$ that maximize the expression ...
0
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0answers
60 views

Best s-term approximation and unit balls in weak $\ell^p$ norm

In the book "A Mathematical Introduction to Compressive Sensing" by Foucart and Rauhut there is the following asymptotic estimate at page 332, equation (11.1): $$\sup_{\mathbf{x}\in B^N_{r,\infty}} ...
1
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1answer
83 views

Nonlinear system of equations whereas most of the equations are linear. How to minimise operation?

Let us say we have a n * n system of equations like KU=F where K is a n*n matrix and U and F are n*1 vectors. K and F are defined and the final goal is to find U values. K is a sparse banded matrix ...
0
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0answers
34 views

How can be a conservative field constraint be efficiently implemented in a continuous optimization problem?

Suppose we have the following continuous optimization problem: $$ \underset{x}{\mathrm{minimize}}f\left(x\right) $$ subject to $$ \exists X:\nabla X=Jac\left(X\right)=x $$ where $f$ is a function ...
-3
votes
1answer
59 views

Convert constraint to do convex optimization or use Lagrange multiplier method [closed]

$w_1, w_2, w_3 ... w_n$ are the weights I need to find I have the following constraint: $|w_1| + |w_2| + .. |w_n| <= 5$ That is the sum of the absolute values of the weights has to be less than ...
4
votes
2answers
276 views

Convex Sets and Nearest Neighbors

For a set $S \subseteq \mathbb{R}^n$ and a point $x \in \mathbb{R}^n$, let $c_S(x)$ be the point $s \in S$ that minimizes $\|x-s\|$ if such a point exists and is uniquely determined. It is known that ...
3
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0answers
82 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 ...
0
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0answers
27 views

Optimization problem involving an entrywise function

Let $X$ a $n\times p$ real-valued matrix and $Y$ a $p\times q$ real-valued matrix. Let $\phi:\mathbb{R} \to \mathbb{R}$ a function. What is the appropriate way to deal with the following optimization ...
0
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0answers
56 views

Convergence of Coordinate Descent / Alternating directions

My question regards this method http://en.wikipedia.org/wiki/Coordinate_descent, where at each step a function $f$ is minimized along one coordinate axis (or block of coordinates). Assume that $f: ...
1
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0answers
72 views

Maximizing the “uniformity” of a probability measure, with constraints, via path length minimization

Background I want to find a probability measure for a continuous random variable, subject to moment constraints, that is maximally "uniform", as defined below: Definition: Maximally Uniform ...
0
votes
0answers
46 views

minimizing concave function

I am interested in using the algorithm of Harold Benson described in his 1991 paper: "A Branch and Bound-Outer Approximation Algorithm for Concave Minimization over a Convex Set". In the paper, he ...