Operations research, linear programming, control theory, systems theory, optimal control, game theory

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0
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0answers
13 views

Find a TSP tour passing at least one nodes in each set of nodes

Given a graph $G$ and a number of node sets, each consisting a number of nodes in $G$. The question is to find the shortest path passing at least one node in each node set. If each node set consists ...
0
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0answers
22 views

is the minimum envelope of two inrersecting convex functions convex? [on hold]

let C1 and C2 two convex curves representing a cost function and intersecting each other once (say). how to prove that there exists at least one point for which C2 > C1?
0
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0answers
31 views

Maximum entropy to fit Johnson distribution by moments [migrated]

I am trying to fit a johnson SU distribution to my data with the first 4 moments. To identify the most suitable set of johnson parameters I am trying to maximize the entropy function. However, I see ...
9
votes
1answer
240 views

Sard's Theorem For Banach Spaces

Given a smooth map from $\phi: B \rightarrow M$ where $B$ is a Banach Space and $M$ is a finite dimensional smooth manifold (for example, the end point map for a control system), what is the strongest ...
2
votes
2answers
146 views

constant rank theorem for banach spaces

Is there a similar statement to the constant rank theorem for finite dim real smooth manifolds which holds for a smooth map $F:B \rightarrow M$ where $B$ is an infinite (countable) dim Banach space ...
1
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0answers
44 views

Optimization with random matrix

Consider $J$ a random matrix of size $n\times n$ with i.i.d. Gaussian entries $J_{ij} \sim \mathcal{N}(0,\sigma^2/n)$. Let $f(x)=tanh(x)$, and for $x\in\mathbb{R}^n$, $f(x)$ denotes the vector where ...
2
votes
1answer
118 views

Stochastic differential equation associated with an optimal control problem

We know how to find the stochastic differential equation (Hamilton-Jacobi-Bellman equation, HJB) of the control problem where a process $X_t$ is controlled up until it is stopped at a stopping time ...
2
votes
1answer
77 views

Functions that are easy to compare to a norm

Let $X$ be a subset of $\mathbb{R}^d$, let $\|\cdot \|_p$ be a norm with $1\leq p\leq\infty$, and let $f:\mathbb{R}^d\to\mathbb{R}$ be a function. I'm trying to find examples of $X$, $p$, and $f$ for ...
2
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1answer
107 views

Largest subgroup of $SU(n)$ for which the adjoint action preserves specific inner product on $\mathfrak{su}(N)$

Given $A,B \in \mathfrak{su}(n)$ such that $K(A, B)=0$, I am looking for the largest subgroup $H$ of $SU(n)$ for which: $K \left(A, Ad_{U}(B) \right) = 0, \ \ \forall U \in H$ where $K$ is the ...
2
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0answers
35 views

LQR solution when there are linear terms in the cost function?

I am trying to solve the following Bellman Equation: $V(s) =\max_u \left[a'u - (u-s)'Q(u-s) + V(u)\right]$ In the equation above, $s,u,a\in \mathbb R^n$, $Q\in \mathbb R^{n\times n}$ is positive ...
0
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0answers
33 views

Global optimisation of the real part of impedance

I have the following global optimisation problem: $$ \underset{\omega}{\min}-c^{T}\left(\omega^{2}\mathbf{1}+A^{2}\right)^{-1}b $$ where $A$ is a $n \times n$ real matrix, $c$ and $d$ are ...
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}} ...
3
votes
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 ...
10
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0answers
131 views

Dividing a convex region to minimize average distances

Let $C$ be a convex region in the plane with area 1 that contains distinct points $p_1,\dots,p_n$. Say I'd like to divide $C$ into $n$ pieces $C_1,\dots,C_n$, each of area $1/n$, and I'd like to ...
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, ...
5
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0answers
56 views

Finding the optimal mixture of two convex functions

I am trying to find an efficient way to solve the problem $$\min_{p,x_1,x_2} p\cdot f(x_1)+ (1-p) \cdot f(x_2)~~~~~ s.t.\\p\cdot g_1(x_1) + (1-p)\cdot g_2(x_2)\leq 1 \\ 0\leq p \leq 1$$ where ...
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 ...
5
votes
1answer
159 views

The Maximal $\ell_2$ norm of a signed sum of vectors

Suppose we have $n$ vectors in $\mathbb{R}^n.$ Consider the signed sum of these vectors: $$U(s_1,\ldots,s_n)=s_1 v_1+s_2 v_2 + \ldots + s_n v_n$$ where $s_j$'s can only take values of $+1$ or $-1.$ I ...
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
0answers
118 views

Fixed area, largest mass — is there a name?

Let $x\in \mathbb{R}^n$ and let $s_k(x)$ denote the sum of the $k$ largest entries of $x$. The function $s_k(x)$ is well-known to be convex and is often used in optimization, such as ...
0
votes
1answer
87 views

How to minimize this sparse quadratic function?

There is a problem when I'm reading a paper. Equation: $min_p|p-p^*|^2+\alpha |R(p)|^2 + \beta |D(p)-\delta|^2$, where $p, p^*, R(p), D(p), \delta$ are all $M\times N$ matrices, and $p^*, R(), D(), ...
2
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2answers
139 views

Why the 'S' in S-procedure/S-lemma?

The S-procedure (also called as S-lemma) is a technique from V. A. Yakubovich that is used to relax a system of quadratic inequalities to a linear matrix inequality problem. It is used largely in ...
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46 views

Projected Alternating Minimization

Assume that $f(x,y)$ is a non-convex function for $x,y\in \mathbb{R}$. Assume that we want to minimize this function (even locally) with respect to $x$ and $y$ such that $x \in \mathcal{X}$ and $y \in ...
1
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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, ...
1
vote
1answer
76 views

What's the most efficient way to solve this euclidean projection on non-negative affine space constraint?

I've come across this convex optimization problem in my research where I need to project a matrix $X_0$ onto a non-negative affine space constraint and box constraints. Concretely, $X \in ...
1
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0answers
45 views

Theorems on stochastic Lyapunov function

Let $X_n$ be a sequence of random variables such that $$P(X_{n+1} \in A|X_m,x_m,m\leq n)= \int_A p(dw|X_n,x_n)$$ It is called a controlled Markov process. Now, suppose there exist $\epsilon_0$, ...
1
vote
1answer
164 views

finding permutation matrix I which minimizes TRACE( I* C*( I^T)* D) matrix

I have a problem that is really important for my thesis and i am not studding math so i will be very glad if you help me in this case... thanks for your help in advance I want to find permutation ...
1
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1answer
47 views

characterization of a certain closed convex cone

Consider $x_1,\cdots,x_n \in \mathbb{R}^d$, and the closed convex cone in $\mathbb{R}^n$ defined by $$\mathcal{K}(\underline{x}):=\{(\varphi(x_1),\cdots,\varphi(x_n)):\varphi \textrm{ convex on ...
2
votes
1answer
166 views

Necessary conditions for optimality in Banach spaces

Let $X$ denote the non-negative "orthant" of the Banach space $L^2$ (or whatever you call the set of functions in $L^2$ that are non-negative), and let $C$ be a closed, convex subset of $X$. Let $f$ ...
7
votes
2answers
206 views

Removing constraints in convex optimization

Say I have a convex optimization problem of the form $$\min_x f(x) ~~ s.t.\\ g_1(x)\leq0,\\\vdots \\g_n(x)\leq 0$$ with all functions convex. Suppose that $x^*$ is a unique optimizer to my problem and ...
2
votes
1answer
89 views

Standard names and methods for this type of fitting minimization

In material science research, we have come across the following type of problem. Given a m by n matrix A, a m vector b, and error tolerance $\varepsilon$, we want to do this minimization $$\eqalign{ ...
3
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0answers
100 views

This function looks quasiconvex, can't understand why

Suppose that $\mathbf{C}$ is a given matrix with non-negative entries in $\mathbb{R}^{m\times n}$ and $d$ is a given scalar, and let $g(\mathbf{y})$ be defined by ...
2
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0answers
88 views

Copositivity under tensor products

Is there any symmetric real matrix $A$ such that $A^{\otimes n}$ is copositive for all positive integers $n$, but such that A is neither positive semidefinite nor has just non-negative entries? ...
0
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1answer
79 views

Sufficient and necessary condition for BIBO stability

I am looking for a reference for the proof of the next claim: "BIBO—bounded input bounded output—stability. We claim that a necessary and sufficient condition for a system described by a linear, ...
0
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0answers
50 views

numerical solver for stochastic optimal control problems

can any one recommend numerical solver (c/c++ library preferred) for stochastic optimal control problems? For deterministic optimal control I found something like that: ...
3
votes
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
votes
1answer
171 views

Linear programming with exponentially many constraints and variables [closed]

From class, I learnt that problems like traveller salesman have a Linear programming representation with exponentially many constraints. Using method of separation, this problem is solved rather ...
7
votes
3answers
345 views

The distribution of the shortest path through $n$ points

In the big picture, I'd like to know: if I sample $n$ points uniformly at random in the unit square, what is the probability that the shortest path that visits each one of them is very small? More ...
1
vote
2answers
145 views

Combinatorial optimization problem involving infinite spin system

In material science research, I am developing an algorithm to solve an infinite combinatorial optimization problem which I believe is the most natural problem when the system size goes to infinity. ...
1
vote
0answers
119 views

lower bound for difference between max cut and min cut

Let $G=(V,E)$ be a graph on $n$ vertices with edge weights $w=(w_e)_{e\in E}$. Let $M^+$ and $M^-$ be the maximum and the minimum cut values, i.e. ...
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 ...
3
votes
1answer
108 views

A difficult combinatorial optimization problem

Let $\mathcal{J}$ be a closed, bounded, compact, convex set in $\mathbb{R}^L$. (Notations: vector $\mathbf{x}$ is denoted in bold letters and its $i^{th}$ co-ordinate is denoted as $x_i$. ...
2
votes
4answers
570 views

Algorithm to find the “optimal” path in a given graph

Assume that $G=(V,E)$ is an undirected connected graph and that $H: V \to \mathbb R$ is a function that assign at each vertex $v \in V$ its height $H(v)$. Think of the pair $(G,H)$ as an energy ...
0
votes
1answer
73 views

Generalized assignment problem with no integrality gap

Suppose I am solving the generalized assignment problem, so that I am given matrices $U$ and $W$ and a vector $c$ (all three of which have, say, positive entries), and I want to solve ...
0
votes
1answer
201 views

A problem about a matrix norm on $\mathfrak{su}(4)$

Given a fixed $B \in \mathfrak{su}(4)$ is it possible to solve for $F$: $\sigma^{\text{max}}\left(\frac{A}{F(A)} + B \right) = 1$, $\forall A \in \mathfrak{su}(4)$. A theorem in ...
3
votes
1answer
52 views

Kálmán filter with delayed (outdated) information

We have a linear system with observation as follows: $x(t+1)=Ax(t)+Bu(t)+w(t)$ $y(t)=Cx(t)+z(t)$ for given information $y(0\sim t)$, we can construct the dynamics of $\hat{x}(t)$ using the Kálmán ...
2
votes
0answers
137 views

First Variation of Dyson Series/Magnus Expansion

Given the matrix differential equation $\frac{dU_t}{dt}=A_t U_t$ there are at least two ways to write a formal solution. Both the Dyson series: $U_t = \mathcal{T} e^{\int_{0}^{t} A_t dt}$ and the ...
4
votes
1answer
74 views

What are the upper bound and stability conditions for the following simple linear system?

Consider the following linear system $$\dot{x}=\sum\limits_{i=1}^{m}{{{\alpha }_{i}}}\left( t \right)\cdot {{A}_{i}}\cdot x \quad (1) $$ where, $x\in {{\mathbb{R}}^{n}}$ represents the state vector, ...
0
votes
1answer
86 views

equation for geodesics of a right invairant Finsler metric on $SU(n)$ which are parallel to a linear affine distribution

I am looking for an equation analogous to the Euler-Poincare equations for a right invariant Finlser metric except I want the geodesics which are parallel to a linear affine distribution on $SU(n)$. ...
0
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0answers
20 views

Zeros of non-lipschitz functions (when noisy estimates are available only)

Given noisy (martingale difference) of a Lipschitz continuous function $f$ it is known how to compute zeros of it. It is the stochastic approximation approach (by Borkar, Kushner and Yin etc.). Is ...