自由半定规划最优解的经验性质
Empirical properties of optima in free semidefinite programs
摘要：
半定规划是基于线性泛函在由线性矩阵不等式定义的凸集上的优化，即$$L_A(X)=I-A_1X_1-\dots-A_g X_g\succeq0.$$形式的不等式这里的$X_j$是实数，并且解的集合被称为谱面体。当$X_i$是任何大小的对称矩阵$n\times n$时，这些不等式是有意义的，并通过张量积$A_i\otimes X_i$进入公式：$L_A(X)\succeq0$的解集被称为自由谱面体，因为它包含所有大小的矩阵，并且定义的"线性束”与矩阵的大小无关。在这篇文章中，我们报告的经验观察到的性质的优化器优化线性泛函的自由光谱限制到矩阵$X_i$固定大小$n\times n$。我们找到的优化器总是经典的极值点。令人惊讶的是，在许多合理的参数范围内，超过99.9%也是自由极值点。此外，活动约束$\ker(L_A(X^\ell))$的维度大约是我们预期的两倍。另一个独特的模式是关于优化元组$(X_1^\ell,\dots,X_g^\ell)$的简化。本文给出了一种将自由谱面体的元素表示为自由极值点的矩阵凸组合的算法，这些表示满足所需自由极值点数目的一个很低的界。
英文摘要：
Semidefinite programming is based on optimization of linear functionals over convex sets defined by linear matrix inequalities, namely, inequalities of the form $$L_A(X)=I-A_1X_1-\dots-A_g X_g\succeq0.$$ Here the $X_j$ are real numbers and the set of solutions is called a spectrahedron. These inequalities make sense when the $X_i$ are symmetric matrices of any size, $n\times n$, and enter the formula though tensor product $A_i\otimes X_i$: The solution set of $L_A(X)\succeq0$ is called a free spectrahedron since it contains matrices of all sizes and the defining ``linear pencil" is ``free" of the sizes of the matrices. In this article, we report on empirically observed properties of optimizers obtained from optimizing linear functionals over free spectrahedra restricted to matrices $X_i$ of fixed size $n\times n$. The optimizers we find are always classical extreme points. Surprisingly, in many reasonable parameter ranges, over 99.9\% are also free extreme points. Moreover, the dimension of the active constraint, $\ker(L_A(X^\ell))$, is about twice what we expected. Another distinctive pattern regards reducibility of optimizing tuples $(X_1^\ell,\dots,X_g^\ell)$. We give an algorithm for representing elements of a free spectrahedron as matrix convex combinations of free extreme points; these representations satisfy a very low bound on the number of free extreme points neede
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