Necessary optimality criteria in mathematical programming in the presence of differentiability

AbstractWe consider the problem of minimizing a function over a region defined by an arbitrary set, equality constraints, and constraints of the inequality type defined via a convex cone. Under some moderate convexity assumptions on the arbitrary set we develop Optimality criteria of the minimum principle type which generalize the Fritz John Optimality conditions. As a consequence of this result necessary Optimality criteria of the saddle point type drop out. Here convexity requirements on the functions are relaxed to convexity at the point under investigation. We then present the weakest possible constraint qualification which insures positivity of the lagrangian multiplier corresponding to the objective function

An augmented Lagrangian decomposition method for quasi-separable problems in MDO

Several decomposition methods have been proposed for the distributed optimal design of quasi-separable problems encountered in Multidisciplinary Design Optimization (MDO). Some of these methods are known to have numerical convergence difficulties that can be explained theoretically. We propose a new decomposition algorithm for quasi-separable MDO problems. In particular, we propose a decomposed problem formulation based on the augmented Lagrangian penalty function and the block coordinate descent algorithm. The proposed solution algorithm consists of inner and outer loops. In the outer loop, the augmented Lagrangian penalty parameters are updated. In the inner loop, our method alternates between solving an optimization master problem, and solving disciplinary optimization subproblems. The coordinating master problem can be solved analytically; the disciplinary subproblems can be solved using commonly available gradient-based optimization algorithms. The augmented Lagrangian decomposition method is derived such that existing proofs can be used to show convergence of the decomposition algorithm to KKT points of the original problem under mild assumptions. We investigate the numerical performance of the proposed method on two example problems. I

Boundless multiobjective models for cash management

"This is an Accepted Manuscript of an article published by Taylor & Francis in Engineering Economist on 31-05-2018, available online: https://doi.org/10.1080/0013791X.2018.1456596"[EN] Cash management models are usually based on a set of bounds that complicate the selection of the optimal policies due to nonlinearity. We here propose to linearize cash management models to guarantee optimality through linear-quadratic multiobjective compromise programming models. We illustrate our approach through a reformulation of the suboptimal state-of-the-art Gormley-Meade¿s model to achieve optimality. Furthermore, we introduce a much simpler formulation that we call the boundless model that also provides optimal solutions without using bounds. Results from a sensitivity analysis using real data sets from 54 different companies show that our boundless model is highly robust to cash flow prediction errors.Generalitat de Catalunya [2014 SGR 118]; Ministerio de Economia y Competitividad [Collectiveware TIN2015-66863-C2-1-R].Salas-Molina, F.; Rodriguez-Aguilar, JA.; Pla Santamaría, D. (2018). Boundless multiobjective models for cash management. Engineering Economist (Online). 63(4):363-381. https://doi.org/10.1080/0013791X.2018.1456596S363381634Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). Coherent Measures of Risk. Mathematical Finance, 9(3), 203-228. doi:10.1111/1467-9965.00068Baccarin, S. (2009). Optimal impulse control for a multidimensional cash management system with generalized cost functions. European Journal of Operational Research, 196(1), 198-206. doi:10.1016/j.ejor.2008.02.040Ballestero, E., & Romero, C. (1998). 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A mathematical framework for contact detection between quadric and superquadric surfaces

The calculation of the minimum distance between surfaces plays an important role in computational mechanics, namely, in the study of constrained multibody systems where contact forces take part. In this paper, a general rigid contact detection methodology for non-conformal bodies, described by ellipsoidal and superellipsoidal surfaces, is presented. The mathematical framework relies on simple algebraic and differential geometry, vector calculus, and on the C2 continuous implicit representations of the surfaces. The proposed methodology establishes a set of collinear and orthogonal constraints between vectors defining the contacting surfaces that, allied with loci constraints, which are specific to the type of surface being used, formulate the contact problem. This set of non-linear equations is solved numerically with the Newton-Raphson method with Jacobian matrices calculated analytically. The method outputs the coordinates of the pair of points with common normal vector directions and, consequently, the minimum distance between both surfaces. Contrary to other contact detection methodologies, the proposed mathematical framework does not rely on polygonal-based geometries neither on complex non-linear optimization formulations. Furthermore, the methodology is extendable to other surfaces that are (strictly) convex, interact in a non-conformal fashion, present an implicit representation, and that are at least C2 continuous. Two distinct methods for calculating the tangent and binormal vectors to the implicit surfaces are introduced: (i) a method based on the Householder reflection matrix; and (ii) a method based on a square plate rotation mechanism. The first provides a base of three orthogonal vectors, in which one of them is collinear to the surface normal. For the latter, it is shown that, by means of an analogy to the referred mechanism, at least two non-collinear vectors to the normal vector can be determined. Complementarily, several mathematical and computational aspects, regarding the rigid contact detection methodology, are described. The proposed methodology is applied to several case tests involving the contact between different (super)ellipsoidal contact pairs. Numerical results show that the implemented methodology is highly efficient and accurate for ellipsoids and superellipsoids.Fundação para a Ciência e a Tecnologia (FCT

NP-hardness of Deciding Convexity of Quartic Polynomials and Related Problems

We show that unless P=NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can decide whether a multivariate polynomial of degree four (or higher even degree) is globally convex. This solves a problem that has been open since 1992 when N. Z. Shor asked for the complexity of deciding convexity for quartic polynomials. We also prove that deciding strict convexity, strong convexity, quasiconvexity, and pseudoconvexity of polynomials of even degree four or higher is strongly NP-hard. By contrast, we show that quasiconvexity and pseudoconvexity of odd degree polynomials can be decided in polynomial time.Comment: 20 page

A deep cut ellipsoid algorithm for convex programming

This paper proposes a deep cut version of the ellipsoid algorithm for solving a general class of continuous convex programming problems. In each step the algorithm does not require more computational effort to construct these deep cuts than its corresponding central cut version. Rules that prevent some of the numerical instabilities and theoretical drawbacks usually associated with the algorithm are also provided. Moreover, for a large class of convex programs a simple proof of its rate of convergence is given and the relation with previously known results is discussed. Finally some computational results of the deep and central cut version of the algorithm applied to a min—max stochastic queue location problem are reported

A Metaheuristic Framework for Bi-level Programming Problems with Multi-disciplinary Applications

Bi-level programming problems arise in situations when the decision maker has to take into account the responses of the users to his decisions. Several problems arising in engineering and economics can be cast within the bi-level programming framework. The bi-level programming model is also known as a Stackleberg or leader-follower game in which the leader chooses his variables so as to optimise his objective function, taking into account the response of the follower(s) who separately optimise their own objectives, treating the leader’s decisions as exogenous. In this chapter, we present a unified framework fully consistent with the Stackleberg paradigm of bi-level programming that allows for the integration of meta-heuristic algorithms with traditional gradient based optimisation algorithms for the solution of bi-level programming problems. In particular we employ Differential Evolution as the main meta-heuristic in our proposal.We subsequently apply the proposed method (DEBLP) to a range of problems from many fields such as transportation systems management, parameter estimation and game theory. It is demonstrated that DEBLP is a robust and powerful search heuristic for this class of problems characterised by non smoothness and non convexity