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

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2
votes
1answer
83 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
68 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
votes
1answer
93 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
votes
0answers
22 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
votes
0answers
31 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
vote
0answers
27 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
95 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
votes
0answers
112 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
64 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
votes
0answers
54 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
votes
0answers
18 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
155 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
61 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
117 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
84 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
votes
2answers
107 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 ...
0
votes
0answers
41 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
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, ...
1
vote
1answer
72 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
vote
0answers
41 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
159 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
vote
1answer
46 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
164 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
202 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
votes
0answers
98 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
votes
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
votes
1answer
73 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
votes
0answers
47 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
160 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
340 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
142 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
116 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
votes
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
509 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
69 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
191 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
51 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
128 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
72 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
81 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
votes
0answers
19 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 ...
7
votes
2answers
432 views

How to solve such an optimization problem

I encounter the following optimization problem, but I can't solve it. Given $N$ variables satisfying $0 \leq x_1 \leq x_2 \leq x_3 \leq ... \leq x_N \leq 1$ and an integer $K$ no large than $N$, find ...
1
vote
0answers
114 views

An optimization problem for a Markov Chain

Consider a Markov Chain $\{X_n\}$ whose transition probability depends on some parameter $\theta$ ($p_{ij}(\theta)$). Now I want to optimize the following quantity $$\lambda(\theta) = ...
2
votes
0answers
34 views

Linear control systems

Are there some algorithms to compute the error bound of the difference between the original system and the balance system using the Balance truncation method?
0
votes
0answers
54 views

Representation of piecewise functions using variational inequalities

I need some help about how to represent piecewise smooth functions defined over N subintervals by using variational inequalities. From a paper I'm analyzing I get: "If we can find a function ...
1
vote
1answer
161 views

Are there some algorithms to solve the diagonal matrix $X$ to the following matrix equation?

Suppose $X$ is an unknown $m \times m$ diagonal matrix. Given a scalar $0<c<1$, and a matrix $A$ of $m \times m$ size whose entries $0<A_{i,j}<1$. Are there some algorithms to find the ...
2
votes
2answers
260 views

Solving a matrix equation $X=c \cdot AXA' +I$ with a diagonal corrections

I am now struggling to solve the matrix $X \in R^{n \times n}$ in the following equation: $X=c \cdot AXA' - diag(c \cdot AXA')+ I$, where (1) $A \in R^{n \times n}$ is a given matrix whose element ...