arXiv subjects
Publications and source records attributed to Kurt Anstreicher.
We establish new properties of the NLP-Id upper bound for the maxi\-mum-entropy sampling problem (MESP). In particular, we give a detailed look at the concavity of its objective function as a function of the scaling parameter employed for NLP bounds for MESP. This leads to more relaxed choices for the scaling parameter for NLP-Id and even improved upper bounds for MESP.
The constrained generalized maximum-entropy sampling problem (CGMESP) is to select an order-s principal submatrix from an order-n covariance matrix, subject to some linear side constraints, so as to maximize the product of its t greatest eigenvalues, 0 < t <= s <n. GMESP refers to the version with no side constraints. Introduced more than 25 years ago, CGMESP is a natural generalization of two fundamental problems in statistical design theory: (i) constrained maximum-entropy sampling problem (CMESP); (ii) binary D-optimality (D-Opt). In the general case, it can be motivated by a selection problem in the context of principal component analysis (PCA). We present novel non-convex extended variable formulations for CGMESP. Using these formulations as points of departure, we present, first non-convex and then convex, continuous relaxations for CGMESP. We demonstrate many relations between different upper bounds for CGMESP, including upper bounds from the literature and our new upper bounds. We investigate the behavior of our relaxations related to the constraints linking the natural variables with the extended variables. We propose and investigate a generalized scaling technique for bound improvement. In the context of branch-and-bound, we determine the better of two natural branching techniques for fixing variables to zero. Finally, we present numerical experiments illustrating the value of our methods.
We consider quadratic optimization in variables $(x,y)$ where $0\le x\le y$, and $y\in\{0,1\}^n$. Such binary $y$ are commonly refered to as "indicator" or "switching" variables and occur commonly in applications. One approach to such problems is based on representing or approximating the convex hull of the set $\{ (x,xx^T, yy^T) : 0\le x\le y\in\{0,1\}^n\}$. A representation for the case $n=1$ is known and has been widely used. We give an exact representation for the case $n=2$ by starting with a disjunctive representation for the convex hull and then eliminating auxilliary variables and constraints that do not change the projection onto the original variables. An alternative derivation for this representation leads to an appealing conjecture for a simplified representation of the convex hull for $n=2$ when the product term $y_1y_2$ is ignored.