A Framework for Globally Optimizing Mixed-Integer Signomial Programs

Mixed-integer signomial optimization problems have broad applicability in engineering. Extending the Global Mixed-Integer Quadratic Optimizer, GloMIQO (Misener, Floudas in J. Glob. Optim., 2012. doi:10.1007/s10898-012-9874-7), this manuscript documents a computational framework for deterministically addressing mixed-integer signomial optimization problems to ε-global optimality. This framework generalizes the GloMIQO strategies of (1) reformulating user input, (2) detecting special mathematical structure, and (3) globally optimizing the mixed-integer nonconvex program. Novel contributions of this paper include: flattening an expression tree towards term-based data structures; introducing additional nonconvex terms to interlink expressions; integrating a dynamic implementation of the reformulation-linearization technique into the branch-and-cut tree; designing term-based underestimators that specialize relaxation strategies according to variable bounds in the current tree node. Computational results are presented along with comparison of the computational framework to several state-of-the-art solvers. © 2013 Springer Science+Business Media New York

Dynamically generated cutting planes for mixed-integer quadratically constrained quadratic programs and their incorporation into GloMIQO 2

<div><p>The global mixed-integer quadratic optimizer, GloMIQO, addresses mixed-integer quadratically constrained quadratic programs (MIQCQP) to ε-global optimality. This paper documents the branch-and-cut framework integrated into GloMIQO 2. Cutting planes are derived from reformulation–linearization technique equations, convex multivariable terms, αBB convexifications, and low- and high-dimensional edge-concave aggregations. Cuts are based on both individual equations and collections of nonlinear terms in MIQCQP. Novel contributions of this paper include: development of a corollary to Crama's [<i>Concave extensions for nonlinear 0-1 maximization problems</i>, Math. Program. 61 (1993), pp. 53–60] necessary and sufficient condition for the existence of a cut dominating the termwise relaxation of a bilinear expression; algorithmic descriptions for deriving each class of cut; presentation of a branch-and-cut framework integrating the cuts. Computational results are presented along with comparison of the GloMIQO 2 performance to several state-of-the-art solvers.</p></div

Effect of entropy on the dynamics of supercooled liquids: New results from high pressure data

We show that for arbitrary thermodynamic conditions, master curves of the entropy are obtained by expressing S(T,V) as a function of TV^g_G, where T is temperature, V specific volume, and g_G the thermodynamic Gruneisen parameter. A similar scaling is known for structural relaxation times,tau = f(TV^g); however, we find g_G < g. We show herein that this inequality reflects contributions to S(T,V) from processes, such as vibrations and secondary relaxations, that do not directly influence the supercooled dynamics. An approximate method is proposed to remove these contributions, S_0, yielding the relationship tau = f(S-S_0).Comment: 10 pages 7 figure

Implementing Parallel Differential Evolution on Spark

[Abstract] Metaheuristics are gaining increased attention as an efficient way of solving hard global optimization problems. Differential Evolution (DE) is one of the most popular algorithms in that class. However, its application to realistic problems results in excessive computation times. Therefore, several parallel DE schemes have been proposed, most of them focused on traditional parallel programming interfaces and infrastruc- tures. However, with the emergence of Cloud Computing, new program- ming models, like Spark, have appeared to suit with large-scale data processing on clouds. In this paper we investigate the applicability of Spark to develop parallel DE schemes to be executed in a distributed environment. Both the master-slave and the island-based DE schemes usually found in the literature have been implemented using Spark. The speedup and efficiency of all the implementations were evaluated on the Amazon Web Services (AWS) public cloud, concluding that the island- based solution is the best suited to the distributed nature of Spark. It achieves a good speedup versus the serial implementation, and shows a decent scalability when the number of nodes grows.[Resumen] Las metaheurísticas están recibiendo una atención creciente como técnica eficiente en la resolución de problemas difíciles de optimización global. Differential Evolution (DE) es una de las metaheurísticas más populares, sin embargo su aplicación en problemas reales deriva en tiempos de cómputo excesivos. Por ello se han realizado diferentes propuestas para la paralelización del DE, en su mayoría utilizando infraestructuras e interfaces de programación paralela tradicionales. Con la aparición de la computación en la nube también se han propuesto nuevos modelos de programación, como Spark, que permiten manejar el procesamiento de datos a gran escala en la nube. En este artículo investigamos la aplicabilidad de Spark en el desarrollo de implementaciones paralelas del DE para su ejecución en entornos distribuidos. Se han implementado tanto la aproximación master-slave como la basada en islas, que son las más comunes. También se han evaluado la aceleración y la eficiencia de todas las implementaciones usando el cloud público de Amazon (AWS, Amazon Web Services), concluyéndose que la implementación basada en islas es la más adecuada para el esquema de distribución usado por Spark. Esta implementación obtiene una buena aceleración en relación a la implementación serie y muestra una escalabilidad bastante buena cuando el número de nodos aumenta.[Resume] As metaheurísticas están recibindo unha atención a cada vez maior como técnica eficiente na resolución de problemas difíciles de optimización global. Differential Evolution (DE) é unha das metaheurísticas mais populares, ainda que a sua aplicación a problemas reais deriva en tempos de cómputo excesivos. É por iso que se propuxeron diferentes esquemas para a paralelización do DE, na sua maioría utilizando infraestruturas e interfaces de programación paralela tradicionais. Coa aparición da computación na nube tamén se propuxeron novos modelos de programación, como Spark, que permiten manexar o procesamento de datos a grande escala na nube. Neste artigo investigamos a aplicabilidade de Spark no desenvolvimento de implementacións paralelas do DE para a sua execución en contornas distribuidas. Implementáronse tanto a aproximación master-slave como a baseada en illas, que son as mais comúns. Tamén se avaliaron a aceleración e a eficiencia de todas as implementacións usando o cloud público de Amazon (AWS, Amazon Web Services), tirando como conclusión que a implementación baseada en illas é a mais acaida para o esquema de distribución usado por Spark. Esta implementación obtén unha boa aceleración en relación á implementación serie e amosa unha escalabilidade bastante boa cando o número de nos aumenta.Ministerio de Economía y Competitividad; DPI2014-55276-C5-2-RXunta de Galicia; GRC2013/055Xunta de Galicia; R2014/04

SQG-Differential Evolution for difficult optimization problems under a tight function evaluation budget

In the context of industrial engineering, it is important to integrate efficient computational optimization methods in the product development process. Some of the most challenging simulation-based engineering design optimization problems are characterized by: a large number of design variables, the absence of analytical gradients, highly non-linear objectives and a limited function evaluation budget. Although a huge variety of different optimization algorithms is available, the development and selection of efficient algorithms for problems with these industrial relevant characteristics, remains a challenge. In this communication, a hybrid variant of Differential Evolution (DE) is introduced which combines aspects of Stochastic Quasi-Gradient (SQG) methods within the framework of DE, in order to improve optimization efficiency on problems with the previously mentioned characteristics. The performance of the resulting derivative-free algorithm is compared with other state-of-the-art DE variants on 25 commonly used benchmark functions, under tight function evaluation budget constraints of 1000 evaluations. The experimental results indicate that the new algorithm performs excellent on the 'difficult' (high dimensional, multi-modal, inseparable) test functions. The operations used in the proposed mutation scheme, are computationally inexpensive, and can be easily implemented in existing differential evolution variants or other population-based optimization algorithms by a few lines of program code as an non-invasive optional setting. Besides the applicability of the presented algorithm by itself, the described concepts can serve as a useful and interesting addition to the algorithmic operators in the frameworks of heuristics and evolutionary optimization and computing

Positive Semidefiniteness and Positive Definiteness of a Linear Parametric Interval Matrix

We consider a symmetric matrix, the entries of which depend linearly on some parameters. The domains of the parameters are compact real intervals. We investigate the problem of checking whether for each (or some) setting of the parameters, the matrix is positive definite (or positive semidefinite). We state a characterization in the form of equivalent conditions, and also propose some computationally cheap sufficient\,/\,necessary conditions. Our results extend the classical results on positive (semi-)definiteness of interval matrices. They may be useful for checking convexity or non-convexity in global optimization methods based on branch and bound framework and using interval techniques

Border Basis for Polynomial System Solving and Optimization

International audienceWe describe the software package borderbasix dedicated to the computation of border bases and the solutions of polynomial equations. We present the main ingredients of the border basis algorithm and the other methods implemented in this package: numerical solutions from multiplication matrices, real radical computation, polynomial optimization. The implementation parameterized by the coefficient type and the choice function provides a versatile family of tools for polynomial computation with modular arithmetic, floating point arithmetic or rational arithmetic. It relies on linear algebra solvers for dense and sparse matrices for these various types of coefficients. A connection with SDP solvers has been integrated for the combination of relaxation approaches with border basis computation. Extensive benchmarks on typical polynomial systems are reported, which show the very good performance of the tool