Operations research, linear programming, control theory, systems theory, optimal control, game theory
2
votes
1answer
44 views
Explicit formula for an LMI solution
Suppose we have a linear matrix inequality (aka LMI aka spectahedron aka linear matrix pencil):
$$A_{0}+x_{1}A_{1}+x_{2}A_{2}+\ldots+x_{m}A_{m} \succeq 0.$$
(The notation $X \succeq Y$ means that ...
0
votes
0answers
38 views
A minimization problem [migrated]
Define $$L(w,u)=\frac{1}{2}\|w-u\|^2+\beta \|\frac{w}{x}\|,~w,u\in R^n$$
where $$\frac{w}{x}=(\frac{w_1}{x_1},\dots, \frac{w_n}{x_n})$$
$$\|x\|=\sqrt{x_1^2+\cdots+x_n^2}$$
Given $u$, $x$ and $\beta$, ...
3
votes
3answers
126 views
A NICE necessary and sufficient condition on positive semi-definiteness of a matrix with a special structure!
Let
$$
A =
\begin{pmatrix}
\sum_{j\ne 1}a_{1j} & -a_{12} & \cdots & -a_{1n}\\
-a_{21} & \sum_{j\ne 2}a_{2j} & \cdots & -a_{2n}\\
\vdots & \vdots & \ddots & ...
0
votes
0answers
27 views
An example of the Sequential Quadratic Programming (SQP) [migrated]
Given an objective function $f(x,y)=(x+6)^2+(y-10)^2$, we want to minimize the function $f$ under a constrained condition $xy-5\leq0$. Obviously it is a constrained optimization, a good algorithm to ...
4
votes
2answers
48 views
A certain type of constrained Rayleigh-Ritz ratio
Let $\mathbf{A_1}$ and $\mathbf{A_2}$ be two hermitian matrices. Consider the problem
\begin{align}
\max_{\mathbf{u}^H\mathbf{u}=1}~\mathbf{u}^H\mathbf{A}_1\mathbf{u} \\\
...
0
votes
0answers
31 views
Optimization with differential inequality constraint
Consider the closed set $[t_1,t_2]⊂R_{>0}$ and $V(t):[t1,t2]→R_{>0}$ being a continuous and piecewise continuously differentiable function. We want to find a continuously differentiable function ...
0
votes
1answer
159 views
Anyone has Kushner's book “Introduction to stochastic control” 1971? I need a theorem from it
In a paper I'm reading, it refers to Theorem 8, Page 217 of the book
"Introduction to Stochastic Control" H. J. Kushner, New York: Holt, Reinhart, and Winston 1971. Unfortunately I don't have it and ...
0
votes
0answers
42 views
Is the dependence of policies on controls redundant if the utility is only state-dependent?
In discrete-time stochastic dynamic programming it is often assumed that the choice of the next action can depend not only on all previous states, but also on all previous actions. I guess, this is ...
1
vote
0answers
50 views
Copositivity in matrix pencils
Given two square symmetric matrices $A,B$ of the same order, the matrix pencil $P(A,B)$ is the set of linear combinations of $A$ and $B$. Finsler's theorem gives an elegant criterion for $P(A,B)$ to ...
3
votes
0answers
100 views
Are there some numerical test to check if a map is a contraction?
Let's say I have a multivariate function
$$
f:D \to D, D \subset \mathbb R ^n, D \text{ compact},
$$
for which there is no closed form.
That is the only way to evaluate the function is to do it ...
2
votes
0answers
40 views
minimize a cost function with matrix traces
Hi, I have a cost function of the form
$$F(X) = \operatorname{tr}(X'AX)+\operatorname{tr}(X'B),\quad\textrm{ s.t. }X'X=I.$$
$X$ is a $m\times n$ matrix, ($m>n$), with orthonormal columns. $A$ is ...
3
votes
1answer
128 views
Delta-convex functions and inner products
A delta-convex (d.c.) function is one which can be written as the difference of two convex functions.
The space of d.c. functions includes all C2 functions, and is interesting because it allows many ...
1
vote
0answers
132 views
Tools for “infinite-dimensional linear programming”
I was wondering, whether you could point me to some tools with which I could tackle the following "infinite-dimensional linear programming" problem:
Notation:
$a=1,2,\ldots, A$, ...
0
votes
0answers
58 views
How to prove equivalence of two Semidefinite optimization models?
I want to prove that the following two models are equivalence.
Is there any suggestion that how could be proved?
First model:
$$\min ~~ E | y| $$
$$~~~~~~~~~~~~s.t. ~~\langle I, W \rangle = k, $$
...
0
votes
1answer
98 views
solve non-convex quadratic constrained quadratic programming
$\min_{\beta}\beta^{T} A \beta$
$s.t. \ \beta^{T} C \beta=1\ and\ \beta\geqslant 0$
Here $A,C\in \mathbb{R}^{M\times M}$, $\beta \in \mathbb{R}^{M}$
I saw in one paper saying that it could be ...
