Proven Optimality Research Articles

Enhancing Quantum Algorithms for Quadratic Unconstrained Binary Optimization via Integer Programming

To date, research in quantum computation promises potential for outperforming classical heuristics in combinatorial optimization. However, when aiming at provable optimality, one has to rely on classical exact methods like integer programming. State-of-the-art integer programming algorithms can compute strong relaxation bounds even for hard instances, but may have to enumerate a large number of subproblems for determining an optimum solution. If the potential of quantum computing realizes, it can be expected that in particular finding high-quality solutions for hard problems can be done fast. Still, near-future quantum hardware considerably limits the size of treatable problems. In this work, we go one step into integrating the potentials of quantum and classical techniques for combinatorial optimization. We propose a hybrid heuristic for the weighted maximum-cut problem and for quadratic unconstrained binary optimization. The heuristic employs a linear programming relaxation, rendering it well-suited for integration into exact branch-and-cut algorithms. For large instances, we reduce the problem size according to a linear relaxation such that the reduced problem can be handled by quantum machines of limited size. Moreover, we improve the applicability of depth-1 QAOA, a parameterized quantum algorithm, by deriving a parameter estimate for arbitrary instances. We present numerous computational results from real quantum hardware.

A Cross-Entropy Approach to the Domination Problem and Its Variants.

The domination problem and three of its variants (total domination, 2-domination, and secure domination) are considered. These problems have various real-world applications, including error correction codes, ad hoc routing for wireless networks, and social network analysis, among others. However, each of them is NP-hard to solve to provable optimality, making fast heuristics for these problems desirable. There are a wealth of highly developed heuristics and approximation algorithms for the domination problem; however, such heuristics are much less common for variants of the domination problem. We redress this gap in the literature by proposing a novel implementation of the cross-entropy method that can be applied to any sensible variant of domination. We present results from experiments that demonstrate that this approach can produce good results in an efficient manner even for larger graphs and that it works roughly as well for any of the domination variants considered.

A polynomial-time dynamic programming algorithm for an optimal picking problem in automated warehouses

We consider an optimization problem arising when a set of items must be selected and picked up from given locations in an automated storage and retrieval system by a crane of given capacity, minimizing the overall distance traveled. The problem has been classified as open in a recent taxonomy of optimal picking problems in automated warehouses. In this paper, we analyze some non-trivial properties of the problem and we describe a polynomial-time dynamic programming algorithm to solve it to proven optimality.

Linear Lexicographic Optimization and Preferential Bidding System

Some airlines use the preferential bidding system to construct the schedules of their pilots. In this system, the pilots bid on the different activities and the schedules that lexicographically maximize the scores of the pilots according to their seniority are selected. A sequential approach to solve this maximization problem is natural: The problem is first solved with the bids of the most senior pilot, and then it is solved with those of the second most senior without decreasing the score of the most senior, and so on. The literature admits that the structure of the problem somehow imposes such an approach. The problem can be modeled as an integer linear lexicographic program. We propose a new efficient method, which relies on column generation for solving its continuous relaxation and returns proven optimality gaps. To design this column generation, we prove that bounded linear lexicographic programs admit “primal-dual” feasible bases, and we show how to compute such bases efficiently. Another contribution on which our method relies is the extension of standard tools for resource-constrained longest path problems to their lexicographic versions. This is useful in our context because the generation of new columns is modeled as a lexicographic resource-constrained longest path problem. Numerical experiments show that this new method is already able to solve to proven optimality industrial instances provided by Air France, with up to 150 pilots. By adding a last ingredient in the resolution of the longest path problems, which exploits the specificity of the preferential bidding system, the method achieves for these instances computational times that are compatible with operational constraints. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.0372 .

Design of Patient Visit Itineraries in Tandem Systems

Problem definition: Multistage service is common in healthcare. One widely adopted approach to manage patient visits in multistage service is to provide patients with visit itineraries that specify personalized appointment time for each patient at each service stage. We study how to design such visit itineraries. Methodology/results: We develop the first optimization modeling framework to provide each patient with a personalized visit itinerary in a tandem (healthcare) service system. Due to interdependencies among stages, our model loses those elegant properties (e.g., L-convexity and submodularity) often utilized to solve the classic single-stage models. To address these challenges, we develop two original reformulations. One is directly amenable to off-the-shelf optimization software, and the other is a concave minimization problem over a polyhedron shown to have neat structural properties, based on which we develop efficient solution algorithms. In addition to these exact solution approaches, we propose an approximation approach with a provable optimality bound and numerically validated performance to serve as an easy-to-implement heuristic. A case study populated by data from the Dana-Farber Cancer Institute shows that our approach makes a remarkable 28% cost reduction over practice on average. Managerial implications: Common approaches used in practice are based on simple adjustments to schedules generated by single-stage models, often assuming deterministic service times. Whereas such approaches are intuitive and take advantage of existing knowledge of single-stage models, they can lead to significant loss of operational efficiency in managing multistage services. A well-designed patient visit itinerary that carefully addresses the interdependencies among stages can significantly improve patient experience and provider utilization. History: This paper was selected for Fast Track in the M&SOM Journal from the 2022 MSOM Healthcare SIG Conference. Funding: The work of the last two authors was supported in part by the National Natural Science Foundation of China [Grants 71931008 and 72001220]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.0134 .

Optimal Program Synthesis via Abstract Interpretation

We consider the problem of synthesizing programs with numerical constants that optimize a quantitative objective, such as accuracy, over a set of input-output examples. We propose a general framework for optimal synthesis of such programs in a given domain specific language (DSL), with provable optimality guarantees. Our framework enumerates programs in a general search graph, where nodes represent subsets of concrete programs. To improve scalability, it uses A * search in conjunction with a search heuristic based on abstract interpretation; intuitively, this heuristic establishes upper bounds on the value of subtrees in the search graph, enabling the synthesizer to identify and prune subtrees that are provably suboptimal. In addition, we propose a natural strategy for constructing abstract transformers for monotonic semantics, which is a common property for components in DSLs for data classification. Finally, we implement our approach in the context of two such existing DSLs, demonstrating that our algorithm is more scalable than existing optimal synthesizers.

Multilabel Feature Selection via Shared Latent Sublabel Structure and Simultaneous Orthogonal Basis Clustering.

Multilabel feature selection solves the dimension distress of high-dimensional multilabel data by selecting the optimal subset of features. Noisy and incomplete labels of raw multilabel data hinder the acquisition of label-guided information. In existing approaches, mapping the label space to a low-dimensional latent space by semantic decomposition to mitigate label noise is considered an effective strategy. However, the decomposed latent label space contains redundant label information, which misleads the capture of potential label relevance. To eliminate the effect of redundant information on the extraction of latent label correlations, a novel method named SLOFS via shared latent sublabel structure and simultaneous orthogonal basis clustering for multilabel feature selection is proposed. First, a latent orthogonal base structure shared (LOBSS) term is engineered to guide the construction of a redundancy-free latent sublabel space via the separated latent clustering center structure. The LOBSS term simultaneously retains latent sublabel information and latent clustering center structure. Moreover, the structure and relevance information of nonredundant latent sublabels are fully explored. The introduction of graph regularization ensures structural consistency in the data space and latent sublabels, thus helping the feature selection process. SLOFS employs a dynamic sublabel graph to obtain a high-quality sublabel space and uses regularization to constrain label correlations on dynamic sublabel projections. Finally, an effective convergence provable optimization scheme is proposed to solve the SLOFS method. The experimental studies on the 18 datasets demonstrate that the presented method performs consistently better than previous feature selection methods.

The family capacitated vehicle routing problem

In this article, we address the family capacitated vehicle routing problem (F-CVRP), an NP-hard problem that generalizes both the FTSP and the capacitated vehicle routing problem. The F-CVRP has practical application in warehouse management in warehouses with scattered storage. We present several mixed integer linear programming formulations for this problem and establish a theoretical and empirical comparison. We also propose valid inequalities adapted from known routing problems from the literature. Some formulations are solved using a branch-and-cut algorithm, which is tested with a newly generated data set. The computational experiment allows us to identify the instances’ most challenging characteristics and the exact methods’ limitations. Finally, we develop an iterated local search (ILS) algorithm to efficiently obtain feasible solutions for the instances that could not be solved to proven optimality. The ILS algorithm is very efficient and can improve the upper bounds obtained by the exact methods within the set time limit.

A New Family of Route Formulations for Split Delivery Vehicle Routing Problems

We propose a new family of formulations with route-based variables for the split delivery vehicle routing problem with and without time windows. Each formulation in this family is characterized by the maximum number of different demand quantities that can be delivered to a customer during a vehicle visit. As opposed to previous formulations in the literature, the exact delivery quantities are not always explicitly known in this new family. The validity of these formulations is ensured by an exponential set of nonrobust constraints. Additionally, we explore a property of optimal solutions that enables us to determine a minimum delivery quantity based on customer demand and vehicle capacity, and this number is often greater than one. We use this property to reduce the number of possible delivery quantities in our formulations, improving the solution times of the computationally strongest formulation in the family. Furthermore, we propose new variants of nonrobust cutting planes that strengthen the formulations, namely limited-memory subset-row covering inequalities and limited-memory strong k-path inequalities. Finally, we develop a branch-cut-and-price (BCP) algorithm to solve our formulations enriched with the proposed valid inequalities, which resorts to state-of-the-art algorithmic enhancements. We show how to effectively manage the nonrobust cuts when solving the pricing problem that dynamically generates route variables. Numerical results indicate that our formulations and BCP algorithm establish new state-of-the-art results for the variant with time windows, as many benchmark instances with 50 and 100 customers are solved to optimality for the first time. Several instances of the variant without time windows are solved to proven optimality for the first time. Funding: This work was supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grants 306033/2019-4, 313220/2020-4, and 314088/2021-0], the Région Nouvelle Aquitaine, France [Grant AAPR2020A-2020-8601810], the Agence Nationale de la Recherche [Grant ANR-20-CE40-0021-01], the Fundação de Amparo à Pesquisa do Estado de São Paulo [Grants 13/07375-0, 16/01860-1, and 19/23596-2], and the Paraíba State Research Foundation [Grants 261/2020 and 041/2023]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.0085 .

Solving the Multi-Choice Two Dimensional Shelf Strip Packing Problem with Time Windows

In the tool coating field, scheduling of production lines requires solving an optimisation problem which we call the multi-choice two-dimensional shelf strip packing problem with time windows. A set of rectangular items needs to be packed in two stages: items are placed on shelves, which in turn are placed on one of several available strips. Crucially, the item's width depends on the chosen strip and each item is associated with a time window such that items can only be placed on the same shelf if their time windows overlap. In collaboration with an industrial partner, this real-world optimisation problem is tackled in this paper by both exact and heuristic methods. The exact method is an arc-flow-based integer linear programming formulation, solved with the commercial solver CPLEX. Experimental evaluation shows that this approach can solve instances to proven optimality with up to 20 different item sizes. Larger, more realistic instances are solved heuristically by an adaptive large neighbourhood search, using first fit and best fit decreasing approaches as repair heuristics. In this way, we obtain high-quality solutions with a remaining optimality gap below 3.3% for instances with up to 2000 different item sizes. The work reported is due to be incorporated into an end-to-end decision support system with the industrial partner.

Electrophysiological Brain Source Imaging via Combinatorial Search with Provable Optimality

Electrophysiological Source Imaging (ESI) refers to reconstructing the underlying brain source activation from non-invasive Electroencephalography (EEG) and Magnetoencephalography (MEG) measurements on the scalp. Estimating the source locations and their extents is a fundamental tool in clinical and neuroscience applications. However, the estimation is challenging because of the ill-posedness and high coherence in the leadfield matrix as well as the noise in the EEG/MEG data. In this work, we proposed a combinatorial search framework to address the ESI problem with a provable optimality guarantee. Specifically, by exploiting the graph neighborhood information in the brain source space, we converted the ESI problem into a graph search problem and designed a combinatorial search algorithm under the framework of A* to solve it. The proposed algorithm is guaranteed to give an optimal solution to the ESI problem. Experimental results on both synthetic data and real epilepsy EEG data demonstrated that the proposed algorithm could faithfully reconstruct the source activation in the brain.

Optimal Sparse Regression Trees.

Regression trees are one of the oldest forms of AI models, and their predictions can be made without a calculator, which makes them broadly useful, particularly for high-stakes applications. Within the large literature on regression trees, there has been little effort towards full provable optimization, mainly due to the computational hardness of the problem. This work proposes a dynamic-programming-with-bounds approach to the construction of provably-optimal sparse regression trees. We leverage a novel lower bound based on an optimal solution to the k-Means clustering algorithm on one dimensional data. We are often able to find optimal sparse trees in seconds, even for challenging datasets that involve large numbers of samples and highly-correlated features.

Exact solution approaches for the discrete α-neighbor p-center problem.

The discrete -neighbor -center problem (d--CP) is an emerging variant of the classical -center problem which recently got attention in literature. In this problem, we are given a discrete set of points and we need to locate facilities on these points in such a way that the maximum distance between each point where no facility is located and its -closest facility is minimized. The only existing algorithms in literature for solving the d--CP are approximation algorithms and two recently proposed heuristics. In this work, we present two integer programming formulations for the d--CP, together with lifting of inequalities, valid inequalities, inequalities that do not change the optimal objective function value and variable fixing procedures. We provide theoretical results on the strength of the formulations and convergence results for the lower bounds obtained after applying the lifting procedures or the variable fixing procedures in an iterative fashion. Based on our formulations and theoretical results, we develop branch-and-cut (B&C) algorithms, which are further enhanced with a starting heuristic and a primal heuristic. We evaluate the effectiveness of our B&Calgorithms using instances from literature. Our algorithms are able to solve 116 out of 194 instances from literature to proven optimality, with a runtime of under a minute for most of them. By doing so, we also provide improved solution values for 116 instances.

Minimal Leader Selection in General Linear Multi-Agent Systems With Switching Topologies: Leveraging Submodularity Ratio

In multi-agent systems with leader-follower dynamics, choosing a subset of agents as leaders is a critical step in achieving the desired coordination performance. In this study, by considering consensus tracking for general linear multi-agent systems under switching topologies, we address the problem of selecting a minimum-size set of leaders by leveraging the submodularity ratio. First, using the dwell time technique, a criterion is derived to ensure that the states of all agents can converge to a reference trajectory that is directly tracked by each leader. Second, exploiting the derived consensus tracking criterion, the metrics with a structure of the Euclidean distance between specific vectors and the space spanned by an iteratively updated matrix are established to identify a set of leaders, and then the corresponding bound of the submodularity ratio is proposed. Third, combining the derived criterion and the constructed metrics, a leader selection scheme is presented together with three polynomial-time algorithms, and the related provable optimality bound of each algorithm can be obtained by leveraging the proposed bound of the submodularity ratio. Finally, illustrative examples are provided to verify the effectiveness of the proposed leader selection scheme.

Multi-period single-allocation hub location-routing: Models and heuristic solutions

This work investigates the use of time-dependent decisions in the context of hub-location routing. Instead of setting up the entire system at once, a planning horizon partitioned into several periods is considered during which the system is to be phased-in. In addition to installing the hubs, decisions are also to be made concerning the hub-level network, namely, the hub edges to use. The origin-destination flows are assumed to be time-dependent as well as the costs underlying the problem which include, set up costs for hubs and hub edges and variable operational costs at the hubs. A mathematical model is developed for the problem that can be solved up to proven optimality with a general-purpose solver for small instances of the problem. For larger instances, a four-phase matheuristic that combines principles of relax-and-fix, variable neighborhood descent and local branching schemes is proposed. In addition, two variants of the matheuristic have been developed. The above model and methodology are tested using data generated by extending existing hub location instances to our problem. The obtained results are detailed and analyzed in depth. The major conclusion is that by capturing time in the decision-making process, one may find solutions that better hedge against parameter changes throughout time. Furthermore, the overall procedure presented in this work is quite general in the sense that it can be easily adapted to other multi-period decision making problems and different objective functions.

Learning sparse nonlinear dynamics via mixed-integer optimization

Discovering governing equations of complex dynamical systems directly from data is a central problem in scientific machine learning. In recent years, the sparse identification of nonlinear dynamics (SINDy) framework, powered by heuristic sparse regression methods, has become a dominant tool for learning parsimonious models. We propose an exact formulation of the SINDy problem using mixed-integer optimization (MIO-SINDy) to solve the sparsity constrained regression problem to provable optimality in seconds. On a large number of canonical ordinary and partial differential equations, we illustrate the dramatic improvement in our approach in accurate model discovery while being more sample efficient, robust to noise, and flexible in accommodating physical constraints.

Arc-flow formulations for the one-dimensional cutting stock problem with multiple manufacturing modes

In this paper, an integration of the one-dimensional cutting stock problem with an operational problem that arises in the manufacture of concrete poles is studied. Seeing that poles have a steel structure, different thicknesses of steel bars can be used in their manufacture. This variety in combining the materials to produce the structure of the poles is known as alternative production modes or multiple manufacturing modes. The problem considered here has the objective of minimizing the total cost to meet the demand for poles using the different available configurations. This problem has already been introduced in the literature and it has been formulated as an integer programming problem. To solve it, the column generation procedure was used. The contribution of this paper is to reformulate the cutting stock problem with multiple manufacturing modes using arc-flow formulations. Arc-flow formulations are promising tools to model and solve complex combinatorial problems. Computational tests are performed comparing the formulations using instances from the literature, which were generated based on the data from a civil construction plant. The arc-flow formulations increased the number of instances solved to proven optimality and also reduced solution time. Lower and upper bounds are also improved when compared with the solution proposed in the literature.

CliSAT: A new exact algorithm for hard maximum clique problems

Given a graph, the maximum clique problem (MCP) asks for determining a complete subgraph with the largest possible number of vertices. We propose a new exact algorithm, called CliSAT, to solve the MCP to proven optimality. This problem is of fundamental importance in graph theory and combinatorial optimization due to its practical relevance for a wide range of applications. The newly developed exact approach is a combinatorial branch-and-bound algorithm that exploits the state-of-the-art branching scheme enhanced by two new bounding techniques with the goal of reducing the branching tree. The first one is based on graph colouring procedures and partial maximum satisfiability problems arising in the branching scheme. The second one is a filtering phase based on constraint programming and domain propagation techniques. CliSAT is designed for structured MCP instances which are computationally difficult to solve since they are dense and contain many interconnected large cliques. Extensive experiments on hard benchmark instances, as well as new hard instances arising from different applications, show that CliSAT outperforms the state-of-the-art MCP algorithms, in some cases by several orders of magnitude.

Online Selection with Convex Costs

We study a novel online optimization problem, termed online selection with convex costs (OSCC). In OSCC, there is a sequence of items, each with a value that remains unknown before its arrival. At each step when there is a new arrival, we need to make an irrevocable decision in terms of whether to accept this item and take its value, or to reject it. The crux of OSCC is that we must pay for an increasing and convex cost associated with the number of items accepted, namely, it is increasingly more difficult to accommodate additional items. The goal is to develop an online algorithm that accepts/ selects a subset of items, without any prior statistical knowledge of future arrival information, to maximize the social surplus, namely, the sum of the accepted items' values minus the total cost. Our main result is the development of a threshold policy that is logistically-simple and easy to implement, but has provable optimality guarantees among all deterministic online algorithms.

Stronger mixed-integer programming-formulations for order- and rack-sequencing in robotic mobile fulfillment systems

This paper addresses the order- and rack-sequencing problem at a single picking station in the context of robotic mobile fulfillment systems, a warehouse technology typically applied in large distribution centers. Following the parts-to-picker concept, items are stored on movable racks that are lifted and transported by automated guided vehicles from the storage area to picking stations for order-processing. The order-picking process involves two linked decisions: How to sequence the processing of orders and how to sequence the rack visits to supply the picking station with the requested items. We present a novel mixed-integer linear programming formulation achieving stronger linear programming bounds than a previous formulation. Including preprocessing techniques it quickly solves instances of medium-size to proven optimality for the first time in literature. For large real-world instances, we provide a three-stage heuristic solution procedure suitable in a dynamic environment, while providing competitive solutions within a short run time. Computational experiments on a broad set of benchmark instances and a comparative study with approaches from literature verify our results.