Showing papers in "Mathematical Programming in 2010"
TL;DR: The concept of a Markov risk measure is introduced and it is used to formulate risk-averse control problems for two Markov decision models: a finite horizon model and a discounted infinite horizon model.
Abstract: We introduce the concept of a Markov risk measure and we use it to formulate risk-averse control problems for two Markov decision models: a finite horizon model and a discounted infinite horizon model. For both models we derive risk-averse dynamic programming equations and a value iteration method. For the infinite horizon problem we develop a risk-averse policy iteration method and we prove its convergence. We also propose a version of the Newton method to solve a nonsmooth equation arising in the policy iteration method and we prove its global convergence. Finally, we discuss relations to min---max Markov decision models.
459 citations
TL;DR: An error bound for the linear convergence analysis of first-order gradient methods for solving convex optimization problems arising in applications, possibly as approximations of intractable problems.
Abstract: Convex optimization problems arising in applications, possibly as approximations of intractable problems, are often structured and large scale. When the data are noisy, it is of interest to bound the solution error relative to the (unknown) solution of the original noiseless problem. Related to this is an error bound for the linear convergence analysis of first-order gradient methods for solving these problems. Example applications include compressed sensing, variable selection in regression, TV-regularized image denoising, and sensor network localization.
261 citations
TL;DR: Computational results for three applications—quadratic facility location, network design with congestion, and portfolio optimization with buy-in thresholds—that show the power of the reformulation technique are presented.
Abstract: We study mixed integer nonlinear programs (MINLP)s that are driven by a collection of indicator variables where each indicator variable controls a subset of the decision variables. An indicator variable, when it is “turned off”, forces some of the decision variables to assume fixed values, and, when it is “turned on”, forces them to belong to a convex set. Many practical MINLPs contain integer variables of this type. We first study a mixed integer set defined by a single separable quadratic constraint and a collection of variable upper and lower bound constraints, and a convex hull description of this set is derived. We then extend this result to produce an explicit characterization of the convex hull of the union of a point and a bounded convex set defined by analytic functions. Further, we show that for many classes of problems, the convex hull can be expressed via conic quadratic constraints, and thus relaxations can be solved via second-order cone programming. Our work is closely related with the earlier work of Ceria and Soares (Math Program 86:595–614, 1999) as well as recent work by Frangioni and Gentile (Math Program 106:225–236, 2006) and, Akturk et al. (Oper Res Lett 37:187–191, 2009). Finally, we apply our results to develop tight formulations of mixed integer nonlinear programs in which the nonlinear functions are separable and convex and in which indicator variables play an important role. In particular, we present computational results for three applications—quadratic facility location, network design with congestion, and portfolio optimization with buy-in thresholds—that show the power of the reformulation technique.
179 citations
TL;DR: In this work nonlinear non-convex multiobjective bilevel optimization problems are discussed using an optimistic approach and it is shown that the set of feasible points of the upper level function, the so-called induced set, can be expressed as theset of minimal solutions of a multiobjectives optimization problem.
Abstract: In this work nonlinear non-convex multiobjective bilevel optimization problems are discussed using an optimistic approach. It is shown that the set of feasible points of the upper level function, the so-called induced set, can be expressed as the set of minimal solutions of a multiobjective optimization problem. This artificial problem is solved by using a scalarization approach by Pascoletti and Serafini combined with an adaptive parameter control based on sensitivity results for this problem. The bilevel optimization problem is then solved by an iterative process using again sensitivity theorems for exploring the induced set and the whole efficient set is approximated. For the case of bicriteria optimization problems on both levels and for a one dimensional upper level variable, an algorithm is presented for the first time and applied to two problems: a theoretical example and a problem arising in applications.
143 citations
TL;DR: This paper proposes new methods for generating valid inequalities from the equation Y = xxT with the non-convex constraint and uses the convex SDP constraint to derive convex quadratic cuts, and combines both approaches in a cutting plane algorithm.
Abstract: This paper addresses the problem of generating strong convex relaxations of Mixed Integer Quadratically Constrained Programming (MIQCP) problems. MIQCP problems are very difficult because they combine two kinds of non- convexities: integer variables and non-convex quadratic constraints. To produce strong relaxations of MIQCP problems, we use techniques from disjunctive programming and the lift-and-project methodology. In particular, we propose new methods for generating valid inequalities from the equation Y = x x T . We use the non-convex constraint $${ Y - x x^T \preccurlyeq 0}$$ to derive disjunctions of two types. The first ones are directly derived from the eigenvectors of the matrix Y − x x T with positive eigenvalues, the second type of disjunctions are obtained by combining several eigenvectors in order to minimize the width of the disjunction. We also use the convex SDP constraint $${ Y - x x^T \succcurlyeq 0}$$to derive convex quadratic cuts, and we combine both approaches in a cutting plane algorithm. We present computational results to illustrate our findings.
132 citations
TL;DR: A new cut selection criterion for Benders’ cuts is proposed and computationally analyzed and the results show that the new criterion is more robust—and often considerably faster—than the standard ones.
Abstract: A new cut selection criterion for Benders’ cuts is proposed and computationally analyzed. The results show that the new criterion is more robust—and often considerably faster—than the standard ones.
129 citations
TL;DR: A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems and global convergence to an $$varepsilon}$$ -global minimizer of the original problem is proved.
Abstract: A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems. At each outer iteration k the method requires the $${\varepsilon_{k}}$$ -global minimization of the Augmented Lagrangian with simple constraints, where $${\varepsilon_k \to \varepsilon}$$ . Global convergence to an $${\varepsilon}$$ -global minimizer of the original problem is proved. The subproblems are solved using the źBB method. Numerical experiments are presented.
128 citations
TL;DR: This paper proposes polynomial-time approximation algorithms with provable worst-case performance ratios for optimization of a multi-linear tensor function over the Cartesian product of spheres, subject to homogeneous quadratic constraints.
Abstract: In this paper, we consider approximation algorithms for optimizing a generic multi-variate homogeneous polynomial function, subject to homogeneous quadratic constraints. Such optimization models have wide applications, e.g., in signal processing, magnetic resonance imaging (MRI), data training, approximation theory, and portfolio selection. Since polynomial functions are non-convex, the problems under consideration are all NP-hard in general. In this paper we shall focus on polynomial-time approximation algorithms. In particular, we first study optimization of a multi-linear tensor function over the Cartesian product of spheres. We shall propose approximation algorithms for such problem and derive worst-case performance ratios, which are shown to be dependent only on the dimensions of the model. The methods are then extended to optimize a generic multi-variate homogeneous polynomial function with spherical constraint. Likewise, approximation algorithms are proposed with provable approximation performance ratios. Furthermore, the constraint set is relaxed to be an intersection of co-centered ellipsoids; namely, we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin, and propose polynomial-time approximation algorithms with provable worst-case performance ratios. Numerical results are reported, illustrating the effectiveness of the approximation algorithms studied.
109 citations
TL;DR: It is shown that there exists a unique lifting function that yields extreme inequalities when starting from a maximal lattice-free triangle with multiple integer points in the relative interior of one of its sides, or a maximal clustering of triangles with integral vertices and one integer point in therelative interior of each side.
Abstract: Recently Andersen et al. [1], Borozan and Cornuejols [6] and Cornuejols and Margot [9] have characterized the extreme valid inequalities of a mixed integer set consisting of two equations with two free integer variables and non-negative continuous variables. These inequalities are either split cuts or intersection cuts derived using maximal lattice-free convex sets. In order to use these inequalities to obtain cuts from two rows of a general simplex tableau, one approach is to extend the system to include all possible non-negative integer variables (giving the two row mixed-integer infinite-group problem), and to develop lifting functions giving the coefficients of the integer variables in the corresponding inequalities. In this paper, we study the characteristics of these lifting functions. We show that there exists a unique lifting function that yields extreme inequalities when starting from a maximal lattice-free triangle with multiple integer points in the relative interior of one of its sides, or a maximal lattice-free triangle with integral vertices and one integer point in the relative interior of each side. In the other cases (maximal lattice-free triangles with one integer point in the relative interior of each side and non-integral vertices, and maximal lattice-free quadrilaterals), non-unique lifting functions may yield distinct extreme inequalities. For the latter family of triangles, we present sufficient conditions to yield an extreme inequality for the two row mixed-integer infinite-group problem.
108 citations
TL;DR: It is shown that a representation for C can be obtained by utilizing the simplex result for n = 4 in conjunction with a triangulation of the 3-cube, and the result for box constraints generalizes the well-known fact that the RLT constraints generate the convex hull of C.
Abstract: Let $${\mathcal{C}}$$ be the convex hull of points $${{\{{1 \choose x}{1 \choose x}^T \,|\, x\in \mathcal{F}\subset \Re^n\}}}$$. Representing or approximating $${\mathcal{C}}$$is a fundamental problem for global optimization algorithms based on convex relaxations of products of variables. We show that if n ≤ 4 and $${\mathcal{F}}$$is a simplex, then $${\mathcal{C}}$$has a computable representation in terms of matrices X that are doubly nonnegative (positive semidefinite and componentwise nonnegative). We also prove that if n = 2 and $${\mathcal{F}}$$is a box, then $${\mathcal{C}}$$has a representation that combines semidefiniteness with constraints on product terms obtained from the reformulation-linearization technique (RLT). The simplex result generalizes known representations for the convex hull of $${{\{(x_1, x_2, x_1x_2)\,|\, x\in\mathcal{F}\}}}$$when $${\mathcal{F}\subset\Re^2}$$is a triangle, while the result for box constraints generalizes the well-known fact that in this case the RLT constraints generate the convex hull of $${{\{(x_1, x_2, x_1x_2)\,|\, x\in\mathcal{F}\}}}$$. When n = 3 and $${\mathcal{F}}$$is a box, we show that a representation for $${\mathcal{C}}$$can be obtained by utilizing the simplex result for n = 4 in conjunction with a triangulation of the 3-cube.
107 citations
TL;DR: This work presents new exact and heuristic separation algorithms for several classes of inequalities, namely lifted cover, extended cover, weight and lifted pack inequalities, and shows how to improve a recent separation algorithm for the 0-1 knapsack polytope itself.
Abstract: Valid inequalities for 0-1 knapsack polytopes often prove useful when tackling hard 0-1 Linear Programming problems. To generate such inequalities, one needs separation algorithms for them, i.e., routines for detecting when they are violated. We present new exact and heuristic separation algorithms for several classes of inequalities, namely lifted cover, extended cover, weight and lifted pack inequalities. Moreover, we show how to improve a recent separation algorithm for the 0-1 knapsack polytope itself. Extensive computational results, on MIPLIB and OR Library instances, show the strengths and limitations of the inequalities and algorithms considered.
TL;DR: The obtained subdifferential estimates are applied to establishing verifiable conditions for the local Lipschitz continuity of the value functions and deriving necessary optimality conditions in parametric DC infinite programs and their remarkable specifications.
Abstract: The paper concerns the study of new classes of parametric optimization problems of the so-called infinite programming that are generally defined on infinite-dimensional spaces of decision variables and contain, among other constraints, infinitely many inequality constraints. These problems reduce to semi-infinite programs in the case of finite-dimensional spaces of decision variables. We focus on DC infinite programs with objectives given as the difference of convex functions subject to convex inequality constraints. The main results establish efficient upper estimates of certain subdifferentials of (intrinsically nonsmooth) value functions in DC infinite programs based on advanced tools of variational analysis and generalized differentiation. The value/marginal functions and their subdifferential estimates play a crucial role in many aspects of parametric optimization including well-posedness and sensitivity. In this paper we apply the obtained subdifferential estimates to establishing verifiable conditions for the local Lipschitz continuity of the value functions and deriving necessary optimality conditions in parametric DC infinite programs and their remarkable specifications. Finally, we employ the value function approach and the established subdifferential estimates to the study of bilevel finite and infinite programs with convex data on both lower and upper level of hierarchical optimization. The results obtained in the paper are new not only for the classes of infinite programs under consideration but also for their semi-infinite counterparts.
TL;DR: A successful heuristic approach is presented, based on an ILP formulation in which the seat requirement constraints are stated in a “strong” form, derived from the description of the convex hull of the variant of the knapsack polytope arising when the sum of the variables is restricted not to exceed two.
Abstract: We face a real-world train-unit assignment problem for an operator running trains in a regional area. Given a set of timetabled train trips, each with a required number of passenger seats, and a set of train units, each with a given number of available seats, the problem calls for an assignment of the train units to the trips, possibly combining more than one train unit for a given trip, that fulfills the seat requests. With respect to analogous case studies previously faced in the literature, ours is characterized by the fairly large number of distinct train-unit types available (in addition to the fairly large number of trips to be covered). As a result, although there is a wide margin of improvement over the solution used by the practitioners (as our results show), even only finding a solution of the same value is challenging in practice. We present a successful heuristic approach, based on an ILP formulation in which the seat requirement constraints are stated in a “strong” form, derived from the description of the convex hull of the variant of the knapsack polytope arising when the sum of the variables is restricted not to exceed two. Computational results on real-world instances are reported, showing the effectiveness of the proposed approach.
TL;DR: A new method is introduced for solving equality constrained nonlinear optimization problems that does not use a penalty function, nor a filter, and yet can be proved to be globally convergent to first-order stationary points.
Abstract: A new method is introduced for solving equality constrained nonlinear optimization problems. This method does not use a penalty function, nor a filter, and yet can be proved to be globally convergent to first-order stationary points. It uses different trust-regions to cope with the nonlinearities of the objective function and the constraints, and allows inexact SQP steps that do not lie exactly in the nullspace of the local Jacobian. Preliminary numerical experiments on CUTEr problems indicate that the method performs well.
TL;DR: This paper studies the influence of noise on subgradient methods for convex constrained optimization, and extends these results to objective functions that are the sum of a large number of convex functions, in which case an incremental subgradient method can be used.
Abstract: In this paper, we study the influence of noise on subgradient methods for convex constrained optimization. The noise may be due to various sources, and is manifested in inexact computation of the subgradients and function values. Assuming that the noise is deterministic and bounded, we discuss the convergence properties for two cases: the case where the constraint set is compact, and the case where this set need not be compact but the objective function has a sharp set of minima (for example the function is polyhedral). In both cases, using several different stepsize rules, we prove convergence to the optimal value within some tolerance that is given explicitly in terms of the errors. In the first case, the tolerance is nonzero, but in the second case, the optimal value can be obtained exactly, provided the size of the error in the subgradient computation is below some threshold. We then extend these results to objective functions that are the sum of a large number of convex functions, in which case an incremental subgradient method can be used.
TL;DR: A new version of this method, called the multi-phase dynamic constraint aggregation method, is proposed, which essentially adds to the original method a partial pricing strategy that involves multiple phases, yielding a faster average computation time per iteration and fewer iterations.
Abstract: Dynamic constraint aggregation is an iterative method that was recently introduced to speed up the linear relaxation solution process of set partitioning type problems. This speed up is mostly due to the use, at each iteration, of an aggregated problem defined by aggregating disjoint subsets of constraints from the set partitioning model. This aggregation is updated when needed to ensure the exactness of the overall approach. In this paper, we propose a new version of this method, called the multi-phase dynamic constraint aggregation method, which essentially adds to the original method a partial pricing strategy that involves multiple phases. This strategy helps keeping the size of the aggregated problem as small as possible, yielding a faster average computation time per iteration and fewer iterations. We also establish theoretical results that provide some insights explaining the success of the proposed method. Tests on the linear relaxation of simultaneous bus and driver scheduling problems involving up to 2,000 set partitioning constraints show that the partial pricing strategy speeds up the original method by an average factor of 4.5.
TL;DR: The concepts of p-inefficiency and polarity cuts are introduced and the procedure is orders of magnitude faster than any of the existing approaches to solve (PSC), and in many cases can reduce hours of computing time to a fraction of a second.
Abstract: In this paper, we address the following probabilistic version (PSC) of the set covering problem: $${\min\{cx\,|\,{\mathbb P}(Ax \ge \xi) \ge p, x \in \{0, 1\}^N\}}$$where A is a 0-1 matrix, $${\xi}$$is a random 0-1 vector and $${p \in (0,1]}$$is the threshold probability level. We introduce the concepts of p-inefficiency and polarity cuts. While the former is aimed at deriving an equivalent MIP reformulation of (PSC), the latter is used as a strengthening device to obtain a stronger formulation. Simplifications of the MIP model which result when one of the following conditions hold are briefly discussed: A is a balanced matrix, A has the circular ones property, the components of $${\xi}$$are pairwise independent, the distribution function of $${\xi}$$is a stationary distribution or has the disjunctive shattering property. We corroborate our theoretical findings by an extensive computational experiment on a test-bed consisting of almost 10,000 probabilistic instances. This test-bed was created using deterministic instances from the literature and consists of probabilistic variants of the set covering model and capacitated versions of facility location, warehouse location and k-median models. Our computational results show that our procedure is orders of magnitude faster than any of the existing approaches to solve (PSC), and in many cases can reduce hours of computing time to a fraction of a second.
TL;DR: This work proves a stronger superlinear convergence result than the above, assuming the usual second-order sufficient condition only, and obtains a new error bound for Karush–Kuhn–Tucker systems for variational problems that holds under an appropriate second- order condition.
Abstract: The stabilized version of the sequential quadratic programming algorithm (sSQP) had been developed in order to achieve fast convergence despite possible degeneracy of constraints of optimization problems, when the Lagrange multipliers associated to a solution are not unique. Superlinear convergence of sSQP had been previously established under the strong second-order sufficient condition for optimality (without any constraint qualification assumptions). We prove a stronger superlinear convergence result than the above, assuming the usual second-order sufficient condition only. In addition, our analysis is carried out in the more general setting of variational problems, for which we introduce a natural extension of sSQP techniques. In the process, we also obtain a new error bound for Karush–Kuhn–Tucker systems for variational problems that holds under an appropriate second-order condition.
TL;DR: A multiobjective optimization program with a feasible set defined by equality constraints and a generalized inequality constraint is studied, which provides necessary second order optimality conditions and also sufficient conditions via a Fritz John type Lagrange multiplier rule and a set-valued second order directional derivative.
Abstract: We study a multiobjective optimization program with a feasible set defined by equality constraints and a generalized inequality constraint. We suppose that the functions involved are Frechet differentiable and their Frechet derivatives are continuous or stable at the point considered. We provide necessary second order optimality conditions and also sufficient conditions via a Fritz John type Lagrange multiplier rule and a set-valued second order directional derivative, in such a way that our sufficient conditions are close to the necessary conditions. Some consequences are obtained for parabolic directionally differentiable functions and C1,1 functions, in this last case, expressed by means of the second order Clarke subdifferential. Some illustrative examples are also given.
TL;DR: This paper establishes that a certain class of non-strongly regular differential variational inequalities is devoid of the Zeno phenomenon, and is of fundamental significance and the first of its kind.
Abstract: The Zeno phenomenon of a switched dynamical system refers to the infinite number of mode switches in finite time. The absence of this phenomenon is crucial to the numerical simulation of such a system by time-stepping methods and to the understanding of the behavior of the system trajectory. Extending a previous result for a strongly regular differential variational inequality, this paper establishes that a certain class of non-strongly regular differential variational inequalities is devoid of the Zeno phenomenon. The proof involves many supplemental results that are of independent interest. Specialized to a frictional contact problem with local compliance and polygonal friction laws, this non-Zenoness result is of fundamental significance and the first of its kind.
TL;DR: A new preference relation in the space of random variables is introduced, which is called robust stochastic dominance, which develops necessary and sufficient conditions of optimality for such optimization problems in the convex case and the nonconvex case.
Abstract: We introduce a new preference relation in the space of random variables, which we call robust stochastic dominance. We consider stochastic optimization problems where risk-aversion is expressed by a robust stochastic dominance constraint. These are composite semi-infinite optimization problems with constraints on compositions of measures of risk and utility functions. We develop necessary and sufficient conditions of optimality for such optimization problems in the convex case. In the nonconvex case, we derive necessary conditions of optimality under additional smoothness assumptions of some mappings involved in the problem.
TL;DR: In an extension of Newton’s method to generalized equations, the implicit function theorem paradigm is carried further and place it in the framework of a mapping acting from the parameter and the starting point to the set of all associated sequences of Newton's iterates as elements of a sequence space.
Abstract: In an extension of Newton’s method to generalized equations, we carry further the implicit function theorem paradigm and place it in the framework of a mapping acting from the parameter and the starting point to the set of all associated sequences of Newton’s iterates as elements of a sequence space. An inverse function version of this result shows that the strong regularity of the mapping associated with the Newton sequences is equivalent to the strong regularity of the generalized equation mapping.
TL;DR: The utility of the convexification tool is extended to relaxing nonconvex inequalities, which are not naturally disjunctive, by providing sufficient conditions for establishing the conveX extension property over the non-negative orthant.
Abstract: In this paper, we derive a closed-form characterization of the convex hull of a generic nonlinear set, when this convex hull is completely determined by orthogonal restrictions of the original set. Although the tools used in this construction include disjunctive programming and convex extensions, our characterization does not introduce additional variables. We develop and apply a toolbox of results to check the technical assumptions under which this convexification tool can be employed. We demonstrate its applicability in integer programming by providing an alternate derivation of the split cut for mixed-integer polyhedral sets and finding the convex hull of certain mixed/pure-integer bilinear sets. We then extend the utility of the convexification tool to relaxing nonconvex inequalities, which are not naturally disjunctive, by providing sufficient conditions for establishing the convex extension property over the non-negative orthant. We illustrate the utility of this result by deriving the convex hull of a continuous bilinear covering set over the non-negative orthant. Although we illustrate our results primarily on bilinear covering sets, they also apply to more general polynomial covering sets for which they yield new tight relaxations.
TL;DR: Under mild assumptions, the equivalence between the new domain reduction and the iterated version of the standard one, allowing a new interpretation of the latter and a new way of computing it is proved.
Abstract: In this paper we discuss domain reduction strategies for global optimization problems with a nonconvex objective function over a bounded convex feasible region. After introducing a standard domain reduction and its iterated version, we will introduce a new reduction strategy. Under mild assumptions, we will prove the equivalence between the new domain reduction and the iterated version of the standard one, allowing a new interpretation of the latter and a new way of computing it. Finally, we prove that any “reasonable” domain reduction strategy is independent of the order by which variables are processed.
TL;DR: The main goal of this article is to present several new results on the maximality of the composition and of the sum of maximal monotone operators in Banach spaces under weak interiority conditions involving their domains.
Abstract: The main goal of this article is to present several new results on the maximality of the composition and of the sum of maximal monotone operators in Banach spaces under weak interiority conditions involving their domains. Direct applications of our results to the structure of the range and domain of a maximal monotone operator are discussed. The last section of this note studies continuity properties of the duality product between a Banach space X and its dual X* with respect to topologies compatible with the natural duality (X × X*, X* × X).
TL;DR: The need for a systematic study of several promising paradigms in mathematical programming that have gained momentum in recent years but have yet to reach the main stream of the field are elucidated.
Abstract: Celebrating the sixtieth anniversary since the zeroth International Symposium on Mathematical Programming was held in 1949, this paper discusses several promising paradigms in mathematical programming that have gained momentum in recent years but have yet to reach the main stream of the field. These are: competition, dynamics, and hierarchy. The discussion emphasizes the interplay between these paradigms and their connections with existing subfields including disjunctive, equilibrium, and nonlinear programming, and variational inequalities. We will describe the modeling approaches, mathematical formulations, and recent results of these paradigms, and sketch some open mathematical and computational challenges arising from the resulting optimization and equilibrium problems. Our goal is to elucidate the need for a systematic study of these problems and to inspire new research in the field.
TL;DR: The main results of the paper include a theorem containing estimates for the surjection modulus of a “partial composition” of set-valued mappings between metric spaces which contains as a particlar case well-known Milyutin's theorem about additive perturbation of a mapping into a Banach space by a Lipschitz mapping.
Abstract: The main results of the paper include (a) a theorem containing estimates for the surjection modulus of a “partial composition” of set-valued mappings between metric spaces which contains as a particlar case well-known Milyutin’s theorem about additive perturbation of a mapping into a Banach space by a Lipschitz mapping; (b) a “double fixed point” theorem for a couple of mappings, one from X into Y and another from Y to X which implies a fairly general version of the set-valued contraction mapping principle and also a certain (different) version of the first theorem.
TL;DR: A primal–dual approximation algorithm is given for both problems whose approximation ratio attains the integrality gap, thereby showing that it is the best possible primal– dual approximation algorithm.
Abstract: We consider the “flow on paths” versions of Max Flow and Min Cut when we restrict to paths having at most B arcs, and for versions where we allow fractional solutions or require integral solutions. We show that the continuous versions are polynomial even if B is part of the input, but that the integral versions are polynomial only when B ≤ 3. However, when B ≤ 3 we show how to solve the problems using ordinary Max Flow/Min Cut. We also give tight bounds on the integrality gaps between the integral and continuous objective values for both problems, and between the continuous objective values for the bounded-length paths version and the version allowing all paths. We give a primal–dual approximation algorithm for both problems whose approximation ratio attains the integrality gap, thereby showing that it is the best possible primal–dual approximation algorithm.
TL;DR: It is proved that the sequential-merge operator creates a very large family of facet-defining inequalities for high-dimensional infinite group problems using facet- defining inequalities of lower-dimensional group problems.
Abstract: One-dimensional infinite group problems have been extensively studied and have yielded strong cutting planes for mixed integer programs. Although numerical and theoretical studies suggest that group cuts can be significantly improved by considering higher-dimensional groups, there are no known facets for infinite group problems whose dimension is larger than two. In this paper, we introduce an operation that we call sequential-merge. We prove that the sequential-merge operator creates a very large family of facet-defining inequalities for high-dimensional infinite group problems using facet-defining inequalities of lower-dimensional group problems. Further, they exhibit two properties that reflect the benefits of using facets of high-dimensional group problems: they have continuous variables’ coefficients that are not dominated by those of the constituent low-dimensional cuts and they can produce cutting planes that do not belong to the first split closure of MIPs. Further, we introduce a general scheme for generating valid inequalities for lower-dimensional group problems using valid inequalities of higher-dimensional group problems. We present conditions under which this construction generates facet-defining inequalities when applied to sequential-merge inequalities. We show that this procedure yields some two-step MIR inequalities of Dash and Gunluk.
TL;DR: The first generic lifting techniques for deriving strong globally valid cuts for nonlinear programs are introduced, and it is demonstrated that these inequalities cannot be obtained using a single round of traditional integer programming cut-generation techniques applied on a tight reformulation of the problem.
Abstract: In this paper, we introduce the first generic lifting techniques for deriving strong globally valid cuts for nonlinear programs. The theory is geometric and provides insights into lifting-based cut generation procedures, yielding short proofs of earlier results in mixed-integer programming. Using convex extensions, we obtain conditions that allow for sequence-independent lifting in nonlinear settings, paving a way for efficient cut-generation procedures for nonlinear programs. This sequence-independent lifting framework also subsumes the superadditive lifting theory that has been used to generate many general-purpose, strong cuts for integer programs. We specialize our lifting results to derive facet-defining inequalities for mixed-integer bilinear knapsack sets. Finally, we demonstrate the strength of nonlinear lifting by showing that these inequalities cannot be obtained using a single round of traditional integer programming cut-generation techniques applied on a tight reformulation of the problem.