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- W3034173836 abstract "The branch of convex optimization called semidefinite programming is based on matrix inequalities (LMI), namely, inequalities of the form $$ L_A (X) = I-A_1 X_1 - dots - A_g X_g succeq 0. $$ Here the $X_j$ are real numbers and the set of solutions to such an inequality is called a spectrahedron. Such an inequality makes sense when the $X_i$ are symmetric matrices of any size, $n times n$, and enter the formula though tensor product $A_i otimes X_i$ with the $A_i$; The solution set of $L_A (X) succeq 0$ is called a spectrahedron since it contains matrices of all sizes and the defining linear pencil is free of the sizes of the matrices. Linear pencils play a heavy role in the burgeoning area of analysis. In this article, we report on numerically observed properties of the extreme points obtained from optimizing a functional $ell$ over a spectrahedron restricted to matrices $X_i$ of fixed size $n times n$. We generate approximately 7 million cases (using various different $g,A_i,n, ell$) and record properties of the resulting optimizers $X^ell$. Of course, the optimizers we find are always Euclidean extreme points, but surprisingly, in many reasonable parameter ranges, over 99.9 % are also extreme points. Moreover, the dimension of the active constraint, kernel $L_A(X^ell)$, is about twice what we expected. Another distinctive pattern we see regards whether or not the optimizing tuple $(X_1^ell, dots, X_g^ell)$ is reducible, i.e., can be simultaneously block diagonalized. In addition we give an algorithm which may be used to represent a given element of a spectrahedron as a matrix convex combination of its extreme points; the representation produced satisfies a low Caratheodory like bound on the number of extreme points needed." @default.
- W3034173836 created "2020-06-12" @default.
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- W3034173836 date "2020-06-03" @default.
- W3034173836 modified "2023-09-23" @default.
- W3034173836 title "Empirical properties of optima in free semidefinite programs" @default.
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