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Abstract This paper investigates convex quadratic optimization problems involvingnindicator variables, each associated with a continuous variable, particularly focusing on scenarios where the matrixQdefining the quadratic term is positive definite and its sparsity pattern corresponds to the adjacency matrix of a tree graph. We introduce a graph-based dynamic programming algorithm that solves this problem in time and memory complexity of$$\mathcal {O}(n^2)$$ $O\left(n^2\right)$. Central to our algorithm is a precise parametric characterization of the cost function across various nodes of the graph corresponding to distinct variables. Our computational experiments conducted on both synthetic and real-world datasets demonstrate the superior performance of our proposed algorithm compared to existing algorithms and state-of-the-art mixed-integer optimization solvers. An important application of our algorithm is in the real-time inference of Gaussian hidden Markov models from data affected by outlier noise. Using a real on-body accelerometer dataset, we solve instances of this problem with over 30,000 variables in under a minute, and its online variant within milliseconds on a standard computer. A Python implementation of our algorithm is available athttps://github.com/aareshfb/Tree-Parametric-Algorithm.git. 

We study the polyhedral convex hull structure of a mixed-integer set which arises in a class of cardinality-constrained concave submodular minimization problems. This class of problems has an objective function in the form of $f(a^Tx)$, where f is a univariate concave function, a is a non-negative vector, and x is a binary vector of appropriate dimension. Such minimization problems frequently appear in applications that involve risk-aversion or economies of scale. We propose three classes of strong valid linear inequalities for this convex hull and specify their facet conditions when a has two distinct values. We show how to use these inequalities to obtain valid inequalities for general a that contains multiple values. We further provide a complete linear convex hull description for this mixed-integer set when a contains two distinct values and the cardinality constraint upper bound is two. Our computational experiments on the mean-risk optimization problem demonstrate the effectiveness of the proposed inequalities in a branch-and-cut framework. 

Abstract We consider the convex quadratic optimization problem in$$\mathbb {R}^{n}$$ $R^n$with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic number of additional variables consists of an$$(n+1) \times (n+1)$$ $(n+1)\times (n+1)$positive semidefinite constraint (explicitly stated) and linear constraints. In particular, convexification of this class of problems reduces to describing a polyhedral set in an extended formulation. While the vertex representation of this polyhedral set is exponential and an explicit linear inequality description may not be readily available in general, we derive a compact mixed-integer linear formulation whose solutions coincide with the vertices of the polyhedral set. We also give descriptions in the original space of variables: we provide a description based on an infinite number of conic-quadratic inequalities, which are “finitely generated.” In particular, it is possible to characterize whether a given inequality is necessary to describe the convex hull. The new theory presented here unifies several previously established results, and paves the way toward utilizing polyhedral methods to analyze the convex hull of mixed-integer nonlinear sets.