Continuous relaxation of MINLP problems by penalty functions: a practical comparison

A practical comparison of penalty functions for globally solving mixed-integer nonlinear programming (MINLP) problems is presented. The penalty approach relies on the continuous relaxation of the MINLP problem by adding a specific penalty term to the objective function. A new penalty algorithm that addresses simultaneously the reduction of the error tolerances for optimality and feasibility, as well as the reduction of the penalty parameter, is designed. Several penalty terms are tested and different penalty parameter update schemes are analyzed. The continuous nonlinear optimization problem is solved by the deterministic DIRECT optimizer. The numerical experiments show that the quality of the produced solutions are satisfactory and that the selected penalties have different performances in terms of efficiency and robustness.This work has been supported by COMPETE: POCI-01-0145-FEDER-007043 and FCT - Fundação para a Ciência e Tecnologia, within the projects UID/CEC/00319/2013 and UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

Revisiting clustering as matrix factorisation on the Stiefel manifold

International audienceThis paper studies clustering for possibly high dimensional data (e.g. images, time series, gene expression data, and many other settings), and rephrase it as low rank matrix estimation in the PAC-Bayesian framework. Our approach leverages the well known Burer-Monteiro factorisation strategy from large scale optimisation, in the context of low rank estimation. Moreover, our Burer-Monteiro factors are shown to lie on a Stiefel manifold. We propose a new generalized Bayesian estimator for this problem and prove novel prediction bounds for clustering. We also devise a componentwise Langevin sampler on the Stiefel manifold to compute this estimator

Shonan Rotation Averaging: Global Optimality by Surfing SO(p)nSO(p)^n

Shonan Rotation Averaging is a fast, simple, and elegant rotation averaging algorithm that is guaranteed to recover globally optimal solutions under mild assumptions on the measurement noise. Our method employs semidefinite relaxation in order to recover provably globally optimal solutions of the rotation averaging problem. In contrast to prior work, we show how to solve large-scale instances of these relaxations using manifold minimization on (only slightly) higher-dimensional rotation manifolds, re-using existing high-performance (but local) structure-from-motion pipelines. Our method thus preserves the speed and scalability of current SFM methods, while recovering globally optimal solutions.Comment: 30 pages (paper + supplementary material). To appear at the European Conference on Computer Vision (ECCV) 202

Projection methods in conic optimization

There exist efficient algorithms to project a point onto the intersection of a convex cone and an affine subspace. Those conic projections are in turn the work-horse of a range of algorithms in conic optimization, having a variety of applications in science, finance and engineering. This chapter reviews some of these algorithms, emphasizing the so-called regularization algorithms for linear conic optimization, and applications in polynomial optimization. This is a presentation of the material of several recent research articles; we aim here at clarifying the ideas, presenting them in a general framework, and pointing out important techniques

Branch and bound based coordinate search filter algorithm for nonsmooth nonconvex mixed-integer nonlinear programming problems

Publicado em "Computational science and its applications – ICCSA 2014...", ISBN 978-3-319-09128-0. Series "Lecture notes in computer science", ISSN 0302-9743, vol. 8580.A mixed-integer nonlinear programming problem (MINLP) is a problem with continuous and integer variables and at least, one nonlinear function. This kind of problem appears in a wide range of real applications and is very difficult to solve. The difficulties are due to the nonlinearities of the functions in the problem and the integrality restrictions on some variables. When they are nonconvex then they are the most difficult to solve above all. We present a methodology to solve nonsmooth nonconvex MINLP problems based on a branch and bound paradigm and a stochastic strategy. To solve the relaxed subproblems at each node of the branch and bound tree search, an algorithm based on a multistart strategy with a coordinate search filter methodology is implemented. The produced numerical results show the robustness of the proposed methodology.This work has been supported by FCT (Fundação para a Ciência e aTecnologia) in the scope of the projects: PEst-OE/MAT/UI0013/2014 and PEst-OE/EEI/UI0319/2014

Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

Immobile indices and CQ-free optimality criteria for linear copositive programming problems

We consider problems of linear copositive programming where feasible sets consist of vectors for which the quadratic forms induced by the corresponding linear matrix combinations are nonnegative over the nonnegative orthant. Given a linear copositive problem, we define immobile indices of its constraints and a normalized immobile index set. We prove that the normalized immobile index set is either empty or can be represented as a union of a finite number of convex closed bounded polyhedra. We show that the study of the structure of this set and the connected properties of the feasible set permits to obtain new optimality criteria for copositive problems. These criteria do not require the fulfillment of any additional conditions (constraint qualifications or other). An illustrative example shows that the optimality conditions formulated in the paper permit to detect the optimality of feasible solutions for which the known sufficient optimality conditions are not able to do this. We apply the approach based on the notion of immobile indices to obtain new formulations of regularized primal and dual problems which are explicit and guarantee strong duality.publishe

An integrated approach to a combinatorial optimisation problem

Funding: MRC grant MR/S003819/1 and Health Data Research UK, an initiative funded by UK Research and Innovation, Department of Health and Social Care (England) and the devolved administrations, and leading medical research charities.We take inspiration from a problem from the healthcare domain, where patients with several chronic conditions follow different guidelines designed for the individual conditions, and where the aim is to find the best treatment plan for a patient that avoids adverse drug reactions, respects patient’s preferences and prioritises drug efficacy. Each chronic condition guideline can be abstractly described by a directed graph, where each node indicates a treatment step (e.g., a choice in medications or resources) and has a certain duration. The search for the best treatment path is seen as a combinatorial optimisation problem and we show how to select a path across the graphs constrained by a notion of resource compatibility. This notion takes into account interactions between any finite number of resources, and makes it possible to express non-monotonic interactions. Our formalisation also introduces a discrete temporal metric, so as to consider only simultaneous nodes in the optimisation process. We express the formal problem as an SMT problem and provide a correctness proof of the SMT code by exploiting the interplay between SMT solvers and the proof assistant Isabelle/HOL. The problem we consider combines aspects of optimal graph execution and resource allocation, showing how an SMT solver can be an alternative to other approaches which are well-researched in the corresponding domains.Postprin