Stabilized dynamic constraint aggregation for solving set partitioning problems

Dynamic constraint aggregation (DCA) and dual variable stabilization (DVS) are two methods that can reduce the negative impact of degeneracy when solving linear programs. The first uses a projection to reduce the primal space whereas the second acts in the dual space. In this paper, we develop a new method, called stabilized dynamic constraint aggregation (SDCA), that combines DCA and DVS for solving set partitioning problems. It allows to fight degeneracy from both primal and dual perspectives simultaneously. To assess the effectiveness of SDCA, we report computational results obtained for highly degenerate multi-depot vehicle scheduling problem instances solved by column generation. These results indicate that SDCA can reduce the average computational time of the master problem by a factor of up to 7 with respect to the best of the two combined methods. Furthermore, they show that its performance is robust with regard to increasing levels of degeneracy in test problems.     

The value function of a mixed integer program: II

We prove that the gap in optimal value, between a mixed-integer program in rationals and its corresponding linear programming relaxation, is bounded as the right-hand-side is varied. In addition, a variant of value iteration is shown to construct subadditive functions which resolve a pure-integer program when no dual degeneracy occurs. These subadditive functions provide solutions to subadditive dual programs for integer programs which are given here, and for which the values of primal and dual problems are equal.     

Geometric Algorithm for Multiparametric Linear Programming

We propose a novel algorithm for solving multiparametric linear programming problems. Rather than visiting different bases of the associated LP tableau, we follow a geometric approach based on the direct exploration of the parameter space. The resulting algorithm has computational advantages, namely the simplicity of its implementation in a recursive form and an efficient handling of primal and dual degeneracy. Illustrative examples describe the approach throughout the paper. The algorithm is used to solve finite-time constrained optimal control problems for discrete-time linear dynamical systems.     

A dual approach to primal degeneracy

The Revised Primal Simplex algorithm, in its simplest form, has no defence against degeneracy. Various forms of the perturbation method are usually effective, but most offer no guarantee of avoiding all degeneracy, and can lead to numerical difficulties. This paper presents a method that avoids cycling and circling by taking a dual approach.The degenerate subproblem consists of all the original variables, but only the degenerate transformed constraints. The current primal objective, which may be mixed, is used. This subproblem may be solved using the dual simplex algorithm, starting from the current dual infeasible solution, and with a zero dual objective. If the dual algorithm terminates optimally then the whole problem is optimal (subject to primal feasibility). Otherwise the final solution provides a non-basic direction which improves the value of the mixed primal objective and moves away from the degenerate vertex. A purification algorithm then renders the solution basic and further improves the mixed objective.     

An incremental primal–dual method for nonlinear programming with special structure

We propose a new class of incremental primal–dual techniques for solving nonlinear programming problems with special structure. Specifically, the objective functions of the problems are sums of independent nonconvex continuously differentiable terms minimized subject to a set of nonlinear constraints for each term. The technique performs successive primal–dual increments for each decomposition term of the objective function. The primal–dual increments are calculated by performing one Newton step towards the solution of the Karush–Kuhn–Tucker optimality conditions of each subproblem associated with each objective function term. We show that the resulting incremental algorithm is q-linearly convergent under mild assumptions for the original problem.     

Computational Experience with Ill-Posed Problems in Semidefinite Programming

Many theoretical and algorithmic results in semidefinite programming are based on the assumption that Slater's constraint qualification is satisfied for the primal and the associated dual problem. We consider semidefinite problems with zero duality gap for which Slater's condition fails for at least one of the primal and dual problem. We propose a numerically reasonable way of dealing with such semidefinite programs. The new method is based on a standard search direction with damped Newton steps towards primal and dual feasibility.     

Strictly feasible solutions and strict complementarity in multiple objective linear optimization

Recently, Luc defined a dual program for a multiple objective linear program. The dual problem is also a multiple objective linear problem and the weak duality and strong duality theorems for these primal and dual problems have been established. Here, we use these results to prove some relationships between multiple objective linear primal and dual problems. We extend the available results on single objective linear primal and dual problems to multiple objective linear primal and dual problems. Complementary slackness conditions for efficient solutions, and conditions for the existence of weakly efficient solution sets and existence of strictly primal and dual feasible points are established. We show that primal-dual (weakly) efficient solutions satisfying strictly complementary conditions exist. Furthermore, we consider Isermann’s and Kolumban’s dual problems and establish conditions for the existence of strictly primal and dual feasible points. We show the existence of primal-dual feasible points satisfying strictly complementary conditions for Isermann’s dual problem. Also, we give an alternative proof to establish necessary conditions for weakly efficient solutions of multiple objective programs, assuming the Kuhn–Tucker (KT) constraint qualification. We also provide a new condition to ensure the KT constraint qualification.     

Accelerating column generation for variable sized bin-packing problems

In this paper, we study different strategies to stabilize and accelerate the column generation method, when it is applied specifically to the variable sized bin-packing problem, or to its cutting stock counterpart, the multiple length cutting stock problem. Many of the algorithms for these problems discussed in the literature rely on column generation, processes that are known to converge slowly due to primal degeneracy and the excessive oscillations of the dual variables. In the sequel, we introduce new dual-optimal inequalities, and explore the principle of model aggregation as an alternative way of controlling the progress of the dual variables. Two algorithms based on aggregation are proposed. The first one relies on a row aggregated LP, while the second one solves iteratively sequences of doubly aggregated models. Working with these approximations, in the various stages of an iterative solution process, has proven to be an effective way of achieving faster convergence.     

A convergent criss-cross method

Our paper presents a new Criss-Cross method for solving linear programming problems. Starting from a neither primal nor dual feasible solution, we reach an optimal solution in finite number of steps if it exists. If there is no optimal solution, then we show that there is not primal feasible or dual feasible solution, We prove the finiteness of this procedure. Our procedure is not the same as the primal or dual simplex method if we have a primal or dual feasible solution, so we have constructed a quite new procedure for solving linear programming problems.     

Cross decomposition for mixed integer programming

Many methods for solving mixed integer programming problems are based either on primal or on dual decomposition, which yield, respectively, a Benders decomposition algorithm and an implicit enumeration algorithm with bounds computed via Lagrangean relaxation. These methods exploit either the primal or the dual structure of the problem. We propose a new approach, cross decomposition, which allows exploiting simultaneously both structures. The development of the cross decomposition method captures profound relationships between primal and dual decomposition. It is shown that the more constraints can be included in the Langrangean relaxation (provided the duality gap remains zero), the fewer the Benders cuts one may expect to need. If the linear programming relaxation has no duality gap, only one Benders cut is needed to verify optimality.     

Graphical analysis of duality and the Kuhn-Tucker conditions in linear programming

Careful inspection of the geometry of the primal linear programming problem reveals the Kuhn-Tucker conditions as well as the dual. Many of the well-known special cases in duality are also seen from the geometry, as well as the complementary slackness conditions and shadow prices. The latter at demonstrated to differ from the dual variables in situations involving primal degeneracy. Virtually all the special relationships between linear programming and duality theory can be seen from the geometry of the primal and an elementary application of vector analysis.     

Combinatoric classes of the transportation problem and their properties

This paper considers the partitioning of all transportation problem instances into a finite number of equivalent classes. Combinatoric properties of each class, in both primal and dual spaces, are investigated. The problem instances are partitioned into convex regions by degeneracy hyperplanes. Properties of adjacent regions are fully developed. It turns out that adjacency relations are much more complex in the primal space than in the dual space. Finally, algorithms based on the simplex method are classified according to how they behave in each class.     

On the connectedness of optimum-degeneracy graphs

So-called optimum-degeneracy graphs describe the structure of the set of primal and dual feasible bases associated with a degenerate vertex of the feasible solution set of a linear program. The structural properties of these graphs play an important role in determining shadow prices or performing sensitivity analysis in linear programming under degeneracy. We prove that general optimum-degeneracy graphs are connected and that negative optimum-degeneracy graphs are connected under certain conditions.     

Properties of the Central Points in Linear Programming Problems

In this paper we list several useful properties of central points in linear programming problems. We study the logarithmic barrier function, the analytic center and the central path, relating the proximity measures and scaled Euclidean distances defined for the primal and primal–dual problems. We study the Newton centering steps, and show how large the short steps used in path following algorithms can actually be, and what variation can be ensured for the barrier function in each iteration of such methods. We relate the primal and primal–dual Newton centering steps and propose a primal-only path following algorithm for linear programming.     

The generalized alternating path algorithm for transportation problems

A new primal extreme point algorithm for solving capacitated transportation problems is developed in this paper. This algorithm, called the generalized alternating path (GAP) algorithm, is a special purpose method specifically designed to take advantage of the often pervasive primal degeneracy of transportation problems.     

Feasibility and solvability of Lyapunov-type linear programming over symmetric cones

In this paper, we consider the Lyapunov-type linear programming and its dual over symmetric cones. By introducing and characterizing the generalized inverse of Lyapunov operator in Euclidean Jordan algebras, we establish two kinds of Lyapunov-type Farkas’ lemmas to exhibit feasibilities of the corresponding primal and dual programming problems, respectively. As one of the main results, we show that the feasibilities of the primal and dual problems lead to the solvability of the primal problem and zero duality gap under some mild condition. In this case, we obtain that any solution to the pair of primal and dual problems is equivalent to the solution of the corresponding KKT system.     

Efficient algorithms for buffer space allocation

This paper describes efficient algorithms for determining how buffer space should be allocated in a flow line. We analyze two problems: a primal problem, which minimizes total buffer space subject to a production rate constraint; and a dual problem, which maximizes production rate subject to a total buffer space constraint. The dual problem is solved by means of a gradient method, and the primal problem is solved using the dual solution. Numerical results are presented. Profit optimization problems are natural generalizations of the primal and dual problems, and we show how they can be solved using essentially the same algorithms.     

Ergodic,primal convergence in dual subgradient schemes for convex programming,II: the case of inconsistent primal problems

Consider the utilization of a Lagrangian dual method which is convergent for consistent convex optimization problems. When it is used to solve an infeasible optimization problem, its inconsistency will then manifest itself through the divergence of the sequence of dual iterates. Will then the sequence of primal subproblem solutions still yield relevant information regarding the primal program? We answer this question in the affirmative for a convex program and an associated subgradient algorithm for its Lagrange dual. We show that the primal–dual pair of programs corresponding to an associated homogeneous dual function is in turn associated with a saddle-point problem, in which—in the inconsistent case—the primal part amounts to finding a solution in the primal space such that the Euclidean norm of the infeasibility in the relaxed constraints is minimized; the dual part amounts to identifying a feasible steepest ascent direction for the Lagrangian dual function. We present convergence results for a conditional \(\varepsilon \)-subgradient optimization algorithm applied to the Lagrangian dual problem, and the construction of an ergodic sequence of primal subproblem solutions; this composite algorithm yields convergence of the primal–dual sequence to the set of saddle-points of the associated homogeneous Lagrangian function; for linear programs, convergence to the subset in which the primal objective is at minimum is also achieved.     

Generalized Lagrangian duality for nonconvex polynomial programs with polynomial multipliers

In this paper, under the existence of a certificate of nonnegativity of the objective function over the given constraint set, we present saddle-point global optimality conditions and a generalized Lagrangian duality theorem for (not necessarily convex) polynomial optimization problems, where the Lagrange multipliers are polynomials. We show that the nonnegativity certificate together with the archimedean condition guarantees that the values of the Lasserre hierarchy of semidefinite programming (SDP) relaxations of the primal polynomial problem converge asymptotically to the common primal–dual value. We then show that the known regularity conditions that guarantee finite convergence of the Lasserre hierarchy also ensure that the nonnegativity certificate holds and the values of the SDP relaxations converge finitely to the common primal–dual value. Finally, we provide classes of nonconvex polynomial optimization problems for which the Slater condition guarantees the required nonnegativity certificate and the common primal–dual value with constant multipliers and the dual problems can be reformulated as semidefinite programs. These classes include some separable polynomial programs and quadratic optimization problems with quadratic constraints that admit certain hidden convexity. We also give several numerical examples that illustrate our results.     

Dynamic convex duality in constrained utility maximization

ABSTRACT

In this paper, we study a constrained utility maximization problem following the convex duality approach. After formulating the primal and dual problems, we construct the necessary and sufficient conditions for both the primal and dual problems in terms of forward and backward stochastic differential equations (FBSDEs) plus some additional conditions. Such formulation then allows us to explicitly characterize the primal optimal control as a function of the adjoint process coming from the dual FBSDEs in a dynamic fashion and vice versa. We also find that the optimal wealth process coincides with the adjoint process of the dual problem and vice versa. Finally we solve three constrained utility maximization problems, which contrasts the simplicity of the duality approach we propose and the technical complexity of solving the primal problem directly.