Branching strategies for mixed-integer programs containing logical constraints and decomposable structure

Decision-making optimisation problems can include discrete selections, e.g. selecting a route, arranging non-overlapping items or designing a network of items. Branch-and-bound (B&B), a widely applied divide-and-conquer framework, often solves such problems by considering a continuous approximation, e.g. replacing discrete variable domains by a continuous superset. Such approximations weaken the logical relations, e.g. for discrete variables corresponding to Boolean variables. Branching in B&B reintroduces logical relations by dividing the search space. This thesis studies designing B&B branching strategies, i.e. how to divide the search space, for optimisation problems that contain both a logical and a continuous structure. We begin our study with a large-scale, industrially-relevant optimisation problem where the objective consists of machine-learnt gradient-boosted trees (GBTs) and convex penalty functions. GBT functions contain if-then queries which introduces a logical structure to this problem. We propose decomposition-based rigorous bounding strategies and an iterative heuristic that can be embedded into a B&B algorithm. We approach branching with two strategies: a pseudocost initialisation and strong branching that target the structure of GBT and convex penalty aspects of the optimisation objective, respectively. Computational tests show that our B&B approach outperforms state-of-the-art solvers in deriving rigorous bounds on optimality. Our second project investigates how satisfiability modulo theories (SMT) derived unsatisfiable cores may be utilised in a B&B context. Unsatisfiable cores are subsets of constraints that explain an infeasible result. We study two-dimensional bin packing (2BP) and develop a B&B algorithm that branches on SMT unsatisfiable cores. We use the unsatisfiable cores to derive cuts that break 2BP symmetries. Computational results show that our B&B algorithm solves 20% more instances when compared with commercial solvers on the tested instances. Finally, we study convex generalized disjunctive programming (GDP), a framework that supports logical variables and operators. Convex GDP includes disjunctions of mathematical constraints, which motivate branching by partitioning the disjunctions. We investigate separation by branching, i.e. eliminating solutions that prevent rigorous bound improvement, and propose a greedy algorithm for building the branches. We propose three scoring methods for selecting the next branching disjunction. We also analyse how to leverage infeasibility to expedite the B&B search. Computational results show that our scoring methods can reduce the number of explored B&B nodes by an order of magnitude when compared with scoring methods proposed in literature. Our infeasibility analysis further reduces the number of explored nodes.Open Acces

Kombinatorisia algoritmeja graafisten mallien oppimiseen

Graphical models are a framework for representing joint distributions over random variables. By capturing the structure of conditional independencies between the variables, a graphical model can express the distribution in a concise factored form that is often efficient to store and reason about. As constructing graphical models by hand is often infeasible, a lot of work has been devoted to learning them automatically from observational data. Of particular interest is the so-called structure learning problem, of finding a graph that encodes the structure of probabilistic dependencies. Once the learner has decided what constitutes a good fit to the data, the task of finding optimal structures typically involves solving an NP-hard problem of combinatorial optimization. While first algorithms for structure learning thus resorted to local search, there has been a growing interest in solving the problem to a global optimum. Indeed, during the past decade multiple exact algorithms have been proposed that are guaranteed to find optimal structures for the family of Bayesian networks, while first steps have been taken for the family of decomposable graphical models. This thesis presents combinatorial algorithms and analytical results with applications in the structure learning problem. For decomposable models, we present exact algorithms for the so-called full Bayesian approach, which involves not only finding individual structures of good fit but also computing posterior expectations of graph features, either by exact computation or via Monte Carlo methods. For Bayesian networks, we study the empirical hardness of the structure learning problem, with the aim of being able to predict the running time of various structure learning algorithms on a given problem instance. As a result, we obtain a hybrid algorithm that effectively combines the best-case performance of multiple existing techniques. Lastly, we study two combinatorial problems of wider interest with relevance in structure learning. First, we present algorithms for counting linear extensions of partially ordered sets, which is required to correct bias in MCMC methods for sampling Bayesian network structures. Second, we give results in the extremal combinatorics of connected vertex sets, whose number bounds the running time of certain algorithms for structure learning and various other problems.Graafiset mallit ovat todennäköisyysmalleja, jotka esittävät muuttujien välisiä tilastollisia suhteita verkkona. Verkon jokainen solmu vastaa yhtä muuttujaa, ja solmujen väliset kaaret kuvaavat muuttujien välisiä riippuvuuksia. Graafinen esitystapa auttaa havainnollistamaan muuttujien kuvaamaa ilmiötä sekä usein mahdollistaa niiden yhteisjakauman esittämisen tiiviissä ja tehokkaasti käsiteltävässä muodossa. Graafisten mallien rakentaminen käsin on kuitenkin usein kohtuuttoman työlästä, mistä syystä niitä on pyritty oppimaan koneellisesti sovittamalla saatavilla olevaan havaintoaineistoon. Erityisesti verkon rakenteen oppiminen on haastava kombinatorinen ongelma, jota on ratkottu pitkälti likimääräisin menetelmin mutta erityisesti viime aikoina myös eksaktisti. Väitöskirjassa esitellään kombinatorisia algoritmeja rakenneoppimisongelman ratkaisemiseksi sekä kokeellisia ja analyyttisiä tuloksia ongelman vaativuudesta. Niin kutsutuille hajoaville graafisille malleille esitellään eksakti algoritmi, joka mahdollistaa sekä yhden optimaalisen verkon löytämisen että niin kutsutun bayesiläisen lähestymistavan, jossa opitaan jakauma kaikkien mahdollisten verkkojen yli. Jakaumasta voidaan joko poimia verkkoja satunnaisotannalla tai se voidaan tiivistää esimerkiksi verkon jokaisen kaaren marginaalitodennäköisyytenä. Bayes-verkot ovat toinen graafisten mallien perhe, joiden oppimiseen on viime aikoina esitetty useita eksakteja algoritmeja. Yksikään tällä hetkellä tunnettu algoritmi ei ole yksiselitteisesti muita nopeampi, vaan eri algoritmit toimivat nopeasti erityyppisillä syötteillä ja niiden ajoajan tarkka arviointi on osoittautunut vaikeaksi. Työn toisessa vaiheessa tutkitaan kokeellisesti, kuinka tällaisten algoritmien ajoaika riippuu annetusta syötteestä, sekä pyritään ennustamaan ajoaikaa koneoppimismenetelmin nopeimman algoritmin valitsemiseksi. Työn loppuosassa tutkitaan kahta kombinatorista ongelmaa, jotka ovat paitsi yleisesti kiinnostavia myös merkittäviä erityisesti Bayes-verkkojen oppimisessa. Ensimmäinen ongelma käsittelee osittaisjärjestysten lineaaristen laajennusten lukumäärän laskemista, jonka avulla korjataan vääristymiä satunnaisotantaan perustuvassa rakenneoppimisessa. Toinen kysymys koskee niin kutsuttujen yhtenäisten solmujoukkojen suurinta mahdollista lukumäärää, joka antaa ylärajan eräiden rakenneoppimisalgoritmien aikavaativuudelle. Suurimmalle lukumäärälle esitetään ylä- ja alarajoja verkoissa, joissa kunkin solmun naapurien lukumäärä on rajoitettu

Tools and Algorithms for the Construction and Analysis of Systems

This open access book constitutes the proceedings of the 28th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2022, which was held during April 2-7, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 46 full papers and 4 short papers presented in this volume were carefully reviewed and selected from 159 submissions. The proceedings also contain 16 tool papers of the affiliated competition SV-Comp and 1 paper consisting of the competition report. TACAS is a forum for researchers, developers, and users interested in rigorously based tools and algorithms for the construction and analysis of systems. The conference aims to bridge the gaps between different communities with this common interest and to support them in their quest to improve the utility, reliability, exibility, and efficiency of tools and algorithms for building computer-controlled systems

Proceedings of the 22nd Conference on Formal Methods in Computer-Aided Design – FMCAD 2022

The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing

Computer Aided Verification

This open access two-volume set LNCS 13371 and 13372 constitutes the refereed proceedings of the 34rd International Conference on Computer Aided Verification, CAV 2022, which was held in Haifa, Israel, in August 2022. The 40 full papers presented together with 9 tool papers and 2 case studies were carefully reviewed and selected from 209 submissions. The papers were organized in the following topical sections: Part I: Invited papers; formal methods for probabilistic programs; formal methods for neural networks; software Verification and model checking; hyperproperties and security; formal methods for hardware, cyber-physical, and hybrid systems. Part II: Probabilistic techniques; automata and logic; deductive verification and decision procedures; machine learning; synthesis and concurrency. This is an open access book