An improved two-stage optimization-based framework for unequal-areas facility layout

The unequal-areas facility layout problem is concerned with finding the optimal arrangement of a given number of non-overlapping indivisible departments with unequal area requirements within a facility. We present an improved optimization-based framework for efficiently finding competitive solutions for this problem. The framework is based on the combination of two mathematical optimization models. The first model is a nonlinear approximation of the problem that establishes the relative position of the departments within the facility, and the second model is an exact convex optimization formulation of the problem that determines the final layout. Aspect ratio constraints on the departments are taken into account by both models. Our computational results show that the proposed framework is computationally efficient and consistently produces competitive, and often improved, layouts for well-known instances from the literature as well as for new large-scale instances with up to 100 departments.     

A new mixed integer programming formulation for facility layout design using flexible bays

This paper presents a mixed-integer programming formulation to find optimal solutions for the block layout problem with unequal departmental areas arranged in flexible bays. The nonlinear department area constraints are modeled in a continuous plane without using any surrogate constraints. The formulation is extensively tested on problems from the literature.     

Applying the sequence-pair representation to optimal facility layout designs

We present a new formulation for the facility layout problem based on the sequence-pair representation, which is used successfully in VLSI design. By tightening the structure of the problem with this formulation, we have extended the solvable solution space from problems with nine departments to problems with eleven departments.     

The volumetric barrier for convex quadratic constraints

Let where and _i is an n×n positive semidefinite matrix. We prove that the volumetric and combined volumetric-logarithmic barriers for are and self-concordant, respectively. Our analysis uses the semidefinite programming (SDP) representation for the convex quadratic constraints defining , and our earlier results on the volumetric barrier for SDP. The self-concordance results actually hold for a class of SDP problems more general than those corresponding to the SDP representation of .Mathematics Subject Classification (1991):90C25, 90C30     

A convex analysis approach for convex multiplicative programming

Global optimization problems involving the minimization of a product of convex functions on a convex set are addressed in this paper. Elements of convex analysis are used to obtain a suitable representation of the convex multiplicative problem in the outcome space, where its global solution is reduced to the solution of a sequence of quasiconcave minimizations on polytopes. Computational experiments illustrate the performance of the global optimization algorithm proposed.      

Stable zero duality gaps in convex programming: Complete dual characterisations with applications to semidefinite programs

The zero duality gap that underpins the duality theory is one of the central ingredients in optimisation. In convex programming, it means that the optimal values of a given convex program and its associated dual program are equal. It allows, in particular, the development of efficient numerical schemes. However, the zero duality gap property does not always hold even for finite-dimensional problems and it frequently fails for problems with non-polyhedral constraints such as the ones in semidefinite programming problems. Over the years, various criteria have been developed ensuring zero duality gaps for convex programming problems. In the present work, we take a broader view of the zero duality gap property by allowing it to hold for each choice of linear perturbation of the objective function of the given problem. Globalising the property in this way permits us to obtain complete geometric dual characterisations of a stable zero duality gap in terms of epigraphs and conjugate functions. For convex semidefinite programs, we establish necessary and sufficient dual conditions for stable zero duality gaps, as well as for a universal zero duality gap in the sense that the zero duality gap property holds for each choice of constraint right-hand side and convex objective function. Zero duality gap results for second-order cone programming problems are also given. Our approach makes use of elegant conjugate analysis and Fenchel's duality.     

Contour line construction for a new rectangular facility in an existing layout with rectangular departments

In a recent paper, Savas et al. [S. Savas, R. Batta, R. Nagi, Finite-size facility placement in the presence of barriers to rectilinear travel, Operations Research 50 (6) (2002) 1018–1031] consider the optimal placement of a finite-sized facility in the presence of arbitrarily shaped barriers under rectilinear travel. Their model applies to a layout context, since barriers can be thought to be existing departments and the finite-sized facility can be viewed as the new department to be placed. In a layout situation, the existing and new departments are typically rectangular in shape. This is a special case of the Savas et al. paper. However the resultant optimal placement may be infeasible due to practical constraints like aisle locations, electrical connections, etc. Hence there is a need for the development of contour lines, i.e. lines of equal objective function value. With these contour lines constructed, one can place the new facility in the best manner. This paper deals with the problem of constructing contour lines in this context. This contribution can also be viewed as the finite-size extension of the contour line result of Francis [R.L. Francis, Note on the optimum location of new machines in existing plant layouts, Journal of Industrial Engineering 14 (2) (1963) 57–59].     

Invariance and efficiency of convex representations

We consider two notions for the representations of convex cones G-representation and lifted-G-representation. The former represents a convex cone as a slice of another; the latter allows in addition, the usage of auxiliary variables in the representation. We first study the basic properties of these representations. We show that some basic properties of convex cones are invariant under one notion of representation but not the other. In particular, we prove that lifted-G-representation is closed under duality when the representing cone is self-dual. We also prove that strict complementarity of a convex optimization problem in conic form is preserved under G-representations. Then we move to study efficiency measures for representations. We evaluate the representations of homogeneous convex cones based on the “smoothness” of the transformations mapping the central path of the representation to the central path of the represented optimization problem. Research of the first author was supported in part by a grant from the Faculty of Mathematics, University of Waterloo and by a Discovery Grant from NSERC. Research of the second author was supported in part by a Discovery Grant from NSERC and a PREA from Ontario, Canada.     

Convergence of the Lasserre hierarchy of SDP relaxations for convex polynomial programs without compactness

We show that the Lasserre hierarchy of semidefinite programming (SDP) relaxations with a slightly extended quadratic module for convex polynomial optimization problems always converges asymptotically even in the case of non-compact semi-algebraic feasible sets. We then prove that the positive definiteness of the Hessian of the associated Lagrangian at a saddle-point guarantees the finite convergence of the hierarchy. We do this by establishing a new sum-of-squares polynomial representation of convex polynomials over convex semi-algebraic sets.     

Market price-based convex risk measures: A distribution-free optimization approach

In Cont (2006) [1], a convex risk measure was proposed to measure the impact of uncertainty resulting from the misspecification of derivative models. Evaluation of the risk measures was illustrated on finite families of probability measures. In this paper, we consider the case of infinite families of measures that share common moments, e.g. mean and variance for European-style options. We show that the risk measure can still be efficiently evaluated based on semi-infinite programming. Examples are given that illustrate the benefits of evaluating the risk measure with infinite families of measures and shed light on the limitations of considering only finite families of measures.     

A note on strong duality in convex semidefinite optimization: necessary and sufficient conditions

A strong duality which states that the optimal values of the primal convex problem and its Lagrangian dual problem are equal (i.e. zero duality gap) and the dual problem attains its maximum is a corner stone in convex optimization. In particular it plays a major role in the numerical solution as well as the application of convex semidefinite optimization. The strong duality requires a technical condition known as a constraint qualification (CQ). Several CQs which are sufficient for strong duality have been given in the literature. In this note we present new necessary and sufficient CQs for the strong duality in convex semidefinite optimization. These CQs are shown to be sharper forms of the strong conical hull intersection property (CHIP) of the intersecting sets of constraints which has played a critical role in other areas of convex optimization such as constrained approximation and error bounds. Research was partially supported by the Australian Research Council. The author is grateful to the referees for their helpful comments     

A non-interior continuation algorithm for solving the convex feasibility problem

In this paper, a specific class of convex feasibility problems are considered and a non-interior continuation algorithm based on a smoothing function to solve this class of problems is introduced. The proposed algorithm solves at most one system of linear equations at each iteration. Under some weak assumptions, we show that the algorithm is globally linearly and locally quadratically convergent. Preliminary numerical results are also reported, which verify the favorable theoretical properties of the proposed algorithm.     

A new algorithm for minimizing convex functions over convex sets

Let be a convex set for which there is an oracle with the following property. Given any pointz∈ℝ ⁿ the oracle returns a “Yes” ifz∈S; whereas ifz∉S then the oracle returns a “No” together with a hyperplane that separatesz fromS. The feasibility problem is the problem of finding a point inS; the convex optimization problem is the problem of minimizing a convex function overS. We present a new algorithm for the feasibility problem. The notion of a volumetric center of a polytope and a related ellipsoid of maximum volume inscribable in the polytope are central to the algorithm. Our algorithm has a significantly better global convergence rate and time complexity than the ellipsoid algorithm. The algorithm for the feasibility problem easily adapts to the convex optimization problem.     

Accelerating Benders' decomposition for the capacitated facility location problem

Discrete facility location problems are attractive candidates for decomposition procedures since two types of decisions have to be performed: on the one hand the yes/no-decision where to locate the facilities, on the other hand the decision how to allocate the demand to the selected facilities. Nevertheless, Benders' decomposition seems to have a rather slow convergence behaviour when applied for solving location problems. In the following, a procedure will be presented for strengthening the Benders' cuts for the capacitated facility location problem. Computational results show the efficiency of the modified Benders' decomposition algorithm. Furthermore, the paretooptimality of the strengthened Benders' cuts in the sense of [Magnanti and Wong 1990] is shown under a weak assumption.This paper was written when the author was at the Institute for Operations Research, University of St. Gallen, Switzerland, and partly supported by Schweizerischer Nationalfond zur Förderung der wissenschaftlichen Forschung (Grant 12-30140.90).     

A new global optimization approach for convex multiplicative programming

In this paper, by solving the relaxed quasiconcave programming problem in outcome space, a new global optimization algorithm for convex multiplicative programming is presented. Two kinds of techniques are employed to establish the algorithm. The first one is outer approximation technique which is applied to shrink relaxation area of quasiconcave programming problem and to compute appropriate feasible points and to raise the capacity of bounding. And the other one is branch and bound technique which is used to guarantee global optimization. Some numerical results are presented to demonstrate the effectiveness and feasibility of the proposed algorithm.     

Polyhedral studies on the convex recoloring problem

A coloring of the vertices of a graph G is convex if, for each assigned color d, the vertices with color d induce a connected subgraph of G. We address the convex recoloring problem, defined as follows. Given a graph G and a coloring of its vertices, recolor a minimum number of vertices of G, so that the resulting coloring is convex. This problem is known to be NP-hard even when G is a path. We show an integer programming formulation for the weighted version of this problem on arbitrary graphs, and then specialize it for trees. We study the facial structure of the polytope defined as the convex hull of the integer points satisfying the restrictions of the proposed ILP formulation, present several classes of facet-defining inequalities and discuss separation algorithms.