arXiv subjects

Qimeng Yu

Publications and source records attributed to Qimeng Yu.

7 recordsLinked to original sources

arXiv2025

Convexification of classes of mixed-integer sets with L$^\natural$-convexity

L$^\natural$ (natural)-convex functions encompass a large class of nonlinear functions over general integer domains and arise in a wide range of real-world applications. We explore the minimization of L$^\natural$-convex functions, of multiple L$^\natural$-convex functions with common variables, and of a mixed-integer extension of L$^\natural$-convex functions -- functions defined over a mixed-integer domain with properties that resemble L$^\natural$-convexity. For each of these families of minimization problems, we propose valid linear inequalities and provide convex hull descriptions for the corresponding epigraphs. For all classes of proposed inequalities, we discuss their facet conditions, develop exact separation methods, and analyze the complexity of the separation problem. We discover hidden L$^\natural$-convexity in well-known mixed-integer structures in the integer programming literature, namely the (general integer) mixing set and the continuous mixing set. We show that our findings subsume the existing polyhedral results for these sets and establish new results for the multi-capacity variant of the continuous mixing set.

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arXiv2023

Mixed-Integer Programming Approaches to Generalized Submodular Optimization and its Applications

Submodularity is an important concept in integer and combinatorial optimization. A classical submodular set function models the utility of selecting homogenous items from a single ground set, and such selections can be represented by binary variables. In practice, many problem contexts involve choosing heterogenous items from more than one ground set or selecting multiple copies of homogenous items, which call for extensions of submodularity. We refer to the optimization problems associated with such generalized notions of submodularity as Generalized Submodular Optimization (GSO). GSO is found in wide-ranging applications, including infrastructure design, healthcare, online marketing, and machine learning. Due to the often highly nonlinear (even non-convex and non-concave) objective function and the mixed-integer decision space, GSO is a broad subclass of challenging mixed-integer nonlinear programming problems. In this tutorial, we first provide an overview of classical submodularity. Then we introduce two subclasses of GSO, for which we present polyhedral theory for the mixed-integer set structures that arise from these problem classes. Our theoretical results lead to efficient and versatile exact solution methods that demonstrate their effectiveness in practical problems using real-world datasets.

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arXiv2022

On Constrained Mixed-Integer DR-Submodular Minimization

DR-submodular functions encompass a broad class of functions which are generally non-convex and non-concave. We study the problem of minimizing any DR-submodular function, with continuous and general integer variables, under box constraints and possibly additional monotonicity constraints. We propose valid linear inequalities for the epigraph of any DR-submodular function under the constraints. We further provide the complete convex hull of such an epigraph, which, surprisingly, turns out to be polyhedral. We propose a polynomial-time exact separation algorithm for our proposed valid inequalities, with which we first establish the polynomial-time solvability of this class of mixed-integer nonlinear optimization problems.

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arXiv2021

Application of integral invariants to apictorial jigsaw puzzle assembly

We present a method for the automatic assembly of apictorial jigsaw puzzles. This method relies on integral area invariants for shape matching and an optimization process to aggregate shape matches into a final puzzle assembly. Assumptions about individual piece shape or arrangement are not necessary. We illustrate our method by solving example puzzles of various shapes and sizes.

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arXiv2021

Strong valid inequalities for a class of concave submodular minimization problems under cardinality constraints

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 $f(a^\top x)$, 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.

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arXiv2020

An Exact Cutting Plane Method for $k$-submodular Function Maximization

A natural and important generalization of submodularity -- $k$-submodularity -- applies to set functions with $k$ arguments and appears in a broad range of applications, such as infrastructure design, machine learning, and healthcare. In this paper, we study maximization problems with $k$-submodular objective functions. We propose valid linear inequalities, namely the $k$-submodular inequalities, for the hypograph of any $k$-submodular function. This class of inequalities serves as a novel generalization of the well-known submodular inequalities. We show that maximizing a $k$-submodular function is equivalent to solving a mixed-integer linear program with exponentially many $k$-submodular inequalities. Using this representation in a delayed constraint generation framework, we design the first exact algorithm, that is not a complete enumeration method, to solve general $k$-submodular maximization problems. Our computational experiments on the multi-type sensor placement problems demonstrate the efficiency of our algorithm in constrained nonlinear $k$-submodular maximization problems for which no alternative compact mixed-integer linear formulations are available. The computational experiments show that our algorithm significantly outperforms the only available exact solution method -- exhaustive search. Problems that would require over 13 years to solve by exhaustive search can be solved within ten minutes using our method.

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arXiv2020

A Polyhedral Approach to Bisubmodular Function Minimization

We consider minimization problems with bisubmodular objective functions. We propose valid inequalities, namely the poly-bimatroid inequalities, and provide a complete linear description of the convex hull of the epigraph of a bisubmodular function. Furthermore, we develop a cutting plane algorithm for constrained bisubmodular minimization based on the poly-bimatroid inequalities. Our computational experiments on the minimization subproblem in robust coupled sensor placement show that our algorithm can solve highly non-linear problems that do not admit compact mixed-integer linear formulations.

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