Using Functional Programming to recognize Named Structure in an Optimization Problem: Application to Pooling

Branch-and-cut optimization solvers typically apply generic algorithms, e.g., cutting planes or primal heuristics, to expedite performance for many mathematical optimization problems. But solver software receives an input optimization problem as vectors of equations and constraints containing no structural information. This article proposes automatically detecting named special structure using the pattern matching features of functional programming. Specifically, we deduce the industrially-relevant nonconvex nonlinear Pooling Problem within a mixed-integer nonlinear optimization problem and show that we can uncover pooling structure in optimization problems which are not pooling problems. Previous work has shown that preprocessing heuristics can find network structures; we show that we can additionally detect nonlinear pooling patterns. Finding named structures allows us to apply, to generic optimization problems, cutting planes or primal heuristics developed for the named structure. To demonstrate the recognition algorithm, we use the recognized structure to apply primal heuristics to a test set of standard pooling problems

Global solution of non-convex quadratically constrained quadratic programs

International audienceThe class of mixed-integer quadratically constrained quadratic programs (QCQP) consists of minimizing a quadratic function under quadratic constraints where the variables could be integer or continuous. On a previous paper we introduced a method called MIQCR for solving QC-QPs with the following restriction : all quadratic sub-functions of purely continuous variables are already convex. In this paper, we propose an extension of MIQCR which applies to any QCQP. Let (P) be a QCQP. Our approach to solve (P) is first to build an equivalent mixed-integer quadratic problem (P *). This equivalent problem (P *) has a quadratic convex objective function, linear constraints, and additional variables y that are meant to satisfy the additional quadratic constraints y = xx T , where x are the initial variables of problem (P). We then propose to solve (P *) by a branch-and-bound algorithm based on the relaxation of the additional quadratic constraints and of the integrality constraints. This type of branching is known as spatial branch-and-bound. Computational experiences are carried out on a total of 325 instances. The results show that the solution time of most of the considered instances is improved by our method in comparison with the recent implementation of QuadProgBB, and with the solvers Cplex, Couenne, Scip, BARON and GloMIQO

Integrality of Linearizations of Polynomials over Binary Variables using Additional Monomials

Polynomial optimization problems over binary variables can be expressed as integer programs using a linearization with extra monomials in addition to those arising in the given polynomial. We characterize when such a linearization yields an integral relaxation polytope, generalizing work by Del Pia and Khajavirad (SIAM Journal on Optimization, 2018) and Buchheim, Crama and Rodr\'iguez-Heck (European Journal of Operations Research, 2019). We also present an algorithm that finds these extra monomials for a given polynomial to yield an integral relaxation polytope or determines that no such set of extra monomials exists. In the former case, our approach yields an algorithm to solve the given polynomial optimization problem as a compact LP, and we complement this with a purely combinatorial algorithm.Comment: 27 pages, 11 figure

Doctor of Philosophy

dissertationIn this dissertation, several problems are solved using different approaches. The first problem is the two-dimensional three-material G-closure problem: finding all possible effective tensors from given conductivities of three materials and volume fractions. We solve this problem by establishing lower bounds on effective tensors, and finding the (sequences of) microstructures that attain the lower bounds. The lower bound is a piece wise analytic function that depends on the conductivity and volume fraction of each component. They are derived using a combination of the translation method, and additional constraints on the field in the materials. The found bound extend their results to the anisotropic case. Furthermore, the lower bound obtained in this dissertation is also the improvement of the Hashin-Shtrikman and translation bounds, in the sense that it is optimal in a range of parameters where previously known bounds are not; and in the region where both the new bound and previously known bounds are not optimal, the bound derived here is tighter. In the case when the established bound is optimal, structures that attain the bound are presented. All structures are laminates of finite rank. While the bound cannot be obtained by laminate structures, we estimate the bound by comparing it with some particular structure. The numerical experiment shows that the gap between the two is rather small, hence the bound is very close to the optimal bound. The next two problems are typical problems in optimal design, and are solved using the variational method in the frame of Young measures developed by Pedregal. The key idea of this approach is to find the quasiconvex envelope of sets and functions. Those ideas have been used before for optimal design problems with two materials at disposal. Our goal here is to explore how those ideas can be extended to three or more materials situations. In particular, we focus on two paradigmatic cases, where we consider a linear-in-the-gradient cost functional and a typical quadratic situation. In both cases, we are able to formulate, quite explicitly, a full relaxation of the problem problem through which optimal microstructures for the original nonconvex problem can be understood. In principle, this approach can be also used to find the G-closure problem as long as one can find the quasiconvex hull of the set, composed of the gradient fields and their associated divergence free fields determined by the governing equations in the G-closure problem

Artificial Intelligence Techniques for Automatic Reformulation and Solution of Structured Mathematical Models

Complex, hierarchical, multi-scale industrial and natural systems generate increasingly large mathematical models. Practitioners are usually able to formulate such models in their "natural" form; however, solving them often requires finding an appropriate reformulation to reveal structures in the model which make it possible to apply efficient, specialized approaches. The search for the "best" formulation of a given problem, the one which allows the application of the solution algorithm that best exploits the available computational resources, is currently a painstaking process which requires considerable work by highly skilled personnel. Experts in solution algorithms are required for figuring out which (formulation, algorithm) pair is better used, considering issues like the appropriate selection of the several obscure algorithmic parameters that each solution methods has. This process is only going to get more complex, as current trends in computer technology dictate the necessity to develop complex parallel approaches capable of harnessing the power of thousands of processing units, thereby adding another layer of complexity in the form of the choice of the appropriate (parallel) architecture. All this renders the use of mathematical models exceedingly costly and difficult for many potentially fruitful applications. The \name{} environment, proposed in this Thesis, aims at devising a software system for automatizing the search for the best combination of (re)formulation, solution algorithm and its parameters (comprised the computational architecture), until now a firm domain of human intervention, to help practitioners bridging the gap between mathematical models cast in their natural form and existing solver systems. I-DARE deals with deep and challenging issues, both from the theoretical and from an implementative viewpoint: 1) the development of a language that can be effectively used to formulate large-scale structured mathematical models and the reformulation rules that allow to transform a formulation into a different one; 2) a core subsystem capable of automatically reformulating the models and searching in the space of (formulations, algorithms, configurations) able to "the best" formulation of a given problem; 3) the design of a general interface for numerical solvers that is capable of accommodate and exploit structure information. To achieve these goals I-DARE will propose a sound and articulated integration of different programming paradigms and techniques like, classic Object-Oriented programing and Artificial Intelligence (Declarative Programming, Frame-Logic, Higher-Order Logic, Machine Learning). By tackling these challenges, I-DARE may have profound, lasting and disruptive effects on many facets of the development and deployment of mathematical models and the corresponding solution algorithms

Computational Multiscale Methods

Computational Multiscale Methods play an important role in many modern computer simulations in material sciences with different time scales and different scales in space. Besides various computational challenges, the meeting brought together various applications from many disciplines and scientists from various scientific communities

Graphical Models with Structured Factors, Neural Factors, and Approximation-aware Training

This thesis broadens the space of rich yet practical models for structured prediction. We introduce a general framework for modeling with four ingredients: (1) latent variables, (2) structural constraints, (3) learned (neural) feature representations of the inputs, and (4) training that takes the approximations made during inference into account. The thesis builds up to this framework through an empirical study of three NLP tasks: semantic role labeling, relation extraction, and dependency parsing -- obtaining state-of-the-art results on the former two. We apply the resulting graphical models with structured and neural factors, and approximation-aware learning to jointly model part-of-speech tags, a syntactic dependency parse, and semantic roles in a low-resource setting where the syntax is unobserved. We present an alternative view of these models as neural networks with a topology inspired by inference on graphical models that encode our intuitions about the data