An efficient algorithm for the Lagrangean dual of nonlinear knapsack problems with additional nested constraints

This paper presents an efficient algorithm for solving the Lagrangean dual of nonlinear knapsack problems with additional nested constraints. The dual solution provides a feasible primal solution (if it exists) and associated lower and upper bounds on the optimal objective function value of the primal problem. Computational experience is cited indicating computation time, number of dual iterations, and “tightness” of the bounds.     

An O(n2) simplex algorithm for a class of linear programs with tree structure

In connection with the optimal design of centralized circuit-free networks linear 0–1 programming problems arise which are related to rooted trees. For each problem the variables correspond to the edges of a given rooted tree T. Each path from a leaf to the root of T, together with edge weights, defines a linear constraint, and a global linear objective is to be maximized. We consider relaxations of such problems where the variables are not restricted to 0 or 1 but are allowed to vary continouosly between these bounds. The values of the optimal solutions of such relaxations may be used in a branch and bound procedure for the original 0–1 problem. While in [10] a primal algorithm for these relaxations is discussed, in this paper, we deal with the dual linear program and present a version of the simplex algorithm for its solution which can be implemented to run in time O(n²). For balanced trees T this time can be reduced to O(n log n).     

Lagrangean relaxation for a lower bound to a set partitioning problem with side constraints: properties and algorithms

The problem (P) addressed here is a special set partitioning problem with two additional non-trivial constraints. A Lagrangean Relaxation (LR_u) is proposed to provide a lower bound to the optimal solution to this problem. This Lagrangean relaxation is accomplished by a subgradient optimization procedure which solves at each iteration a special 0–1 knapsack problem (KP-k). We give two procedures to solve (KP-k), namely an implicity enumeration algorithm and a column generation method. The approach is promising for it provides feasible integer solutions to the side constraints that will hopefully be optimal to most of the instances of the problem (P). Properties of the feasible solutions to (KP-k) are highlighted and it is shown that the linear programming relaxation to this problem has a worst case time bound of order O(n³).     

A one-phase algorithm for semi-infinite linear programming

We present an algorithm for solving a large class of semi-infinite linear programming problems. This algorithm has several advantages: it handles feasibility and optimality together; it has very weak restrictions on the constraints; it allows cuts that are not near the most violated cut; and it solves the primal and the dual problems simultaneously. We prove the convergence of this algorithm in two steps. First, we show that the algorithm can find an-optimal solution after finitely many iterations. Then, we use this result to show that it can find an optimal solution in the limit. We also estimate how good an-optimal solution is compared to an optimal solution and give an upper bound on the total number of iterations needed for finding an-optimal solution under some assumptions. This algorithm is generalized to solve a class of nonlinear semi-infinite programming problems. Applications to convex programming are discussed.     

Using the parametric approach to solve the continuous-time linear fractional max–min problems

A numerical algorithm based on parametric approach is proposed in this paper to solve a class of continuous-time linear fractional max-min programming problems. We shall transform this original problem into a continuous-time non-fractional programming problem, which unfortunately happens to be a continuous-time nonlinear programming problem. In order to tackle this nonlinear problem, we propose the auxiliary problem that will be formulated as a parametric continuous-time linear programming problem. We also introduce a dual problem of this parametric continuous-time linear programming problem in which the weak duality theorem also holds true. We introduce the discrete approximation method to solve the primal and dual pair of parametric continuous-time linear programming problems by using the recurrence method. Finally, we provide two numerical examples to demonstrate the usefulness of this algorithm.     

Using the Dinkelbach-type algorithm to solve the continuous-time linear fractional programming problems

A Dinkelbach-type algorithm is proposed in this paper to solve a class of continuous-time linear fractional programming problems. We shall transform this original problem into a continuous-time non-fractional programming problem, which unfortunately happens to be a continuous-time nonlinear programming problem. In order to tackle this nonlinear problem, we propose the auxiliary problem that will be formulated as parametric continuous-time linear programming problem. We also introduce a dual problem of this parametric continuous-time linear programming problem in which the weak duality theorem also holds true. We introduce the discrete approximation method to solve the primal and dual pair of parametric continuous-time linear programming problems by using the recurrence method. Finally, we provide two numerical examples to demonstrate the usefulness of this practical algorithm.     

An application of gomory cuts in number theory

Frobenius has stated the following problem. Suppose thata ₁, a₂, ?, a_n are given positive integers and g.c.d. (a ₁, ? , a_n) = 1. The problem is to determine the greatest positive integerg so that the equation $$\sum\limits_{i = 1}^n {a_i x_i = g} $$ has no nonnegative integer solution. Showing the interrelation of the original problem and discrete optimization we give lower bounds for this number using Gomory cuts which are tools for solving discrete programming problems. In the first section an important theorem is cited after some remarks. In Section 2 we state a parametric knapsack problem. The Frobenius problem is equivalent with finding the value of the parameter where the optimal objective function value is maximal. The basis of this reformulation is the above mentioned theorem. Gomory's cutting plane method is applied for the knapsack problem in Section 3. Only one cut is generated and we make one dual simplex step after cutting the linear programming optimum of the knapsack problem. Applying this result we gain lower bounds for the Frobenius problem in Section 4. In the last section we show that the bounds are sharp in the sense that there are examples with arbitrary many coefficients where the lower bounds and the exact solution of the Frobenius problem coincide.     

A mixed-integer linear programming problem which is efficiently solvable

Efficient algorithms are known for the simple linear programming problem where each inequality is of the form x_j − x_i ≤ a_ij. Furthermore, these techniques extend to the integer linear programming variant of the problem. This paper gives an efficient solution to the mixed-integer linear programming variant where some, but not necessarily all, of the unknowns are required to be integers. The algorithm we develop is based on a graph representation of the constraint system and runs in O(|V||E| + |V|²lg|V|) time. It has several applications including optimal retiming of synchronous circuitry, VLSI layout compaction in the presence of power and ground buses, and PERT scheduling with periodic constraints.     

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.     

Solving convex quadratic bilevel programming problems using an enumeration sequential quadratic programming algorithm

In this paper, we present an original method to solve convex bilevel programming problems in an optimistic approach. Both upper and lower level objective functions are convex and the feasible region is a polyhedron. The enumeration sequential linear programming algorithm uses primal and dual monotonicity properties of the primal and dual lower level objective functions and constraints within an enumeration frame work. New optimality conditions are given, expressed in terms of tightness of the constraints of lower level problem. These optimality conditions are used at each step of our algorithm to compute an improving rational solution within some indexes of lower level primal-dual variables and monotonicity networks as well. Some preliminary computational results are reported.     

Convergence and polynomiality of primal-dual interior-point algorithms for linear programming with selective addition of inequalities

This paper presents the convergence proof and complexity analysis of an interior-point framework that solves linear programming problems by dynamically selecting and adding relevant inequalities. First, we formulate a new primal–dual interior-point algorithm for solving linear programmes in non-standard form with equality and inequality constraints. The algorithm uses a primal–dual path-following predictor–corrector short-step interior-point method that starts with a reduced problem without any inequalities and selectively adds a given inequality only if it becomes active on the way to optimality. Second, we prove convergence of this algorithm to an optimal solution at which all inequalities are satisfied regardless of whether they have been added by the algorithm or not. We thus provide a theoretical foundation for similar schemes already used in practice. We also establish conditions under which the complexity of such algorithm is polynomial in the problem dimension and address remaining limitations without these conditions for possible further research.     

Exact algorithm over an arc-time-indexed formulation for parallel machine scheduling problems

This paper presents an exact algorithm for the identical parallel machine scheduling problem over a formulation where each variable is indexed by a pair of jobs and a completion time. We show that such a formulation can be handled, in spite of its huge number of variables, through a branch cut and price algorithm enhanced by a number of practical techniques, including a dynamic programming procedure to fix variables by Lagrangean bounds and dual stabilization. The resulting method permits the solution of many instances of the P||∑w _j T _j problem with up to 100 jobs, and having 2 or 4 machines. This is the first time that medium-sized instances of the P||∑w _j T _j have been solved to optimality.     

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.     

A dual algorithm for the one-machine scheduling problem

A branch and bound algorithm is presented for the problem of schedulingn jobs on a single machine to minimize tardiness. The algorithm uses a dual problem to obtain a good feasible solution and an extremely sharp lower bound on the optimal objective value. To derive the dual problem we regard the single machine as imposing a constraint for each time period. A dual variable is associated with each of these constraints and used to form a Lagrangian problem in which the dualized constraints appear in the objective function. A lower bound is obtained by solving the Lagrangian problem with fixed multiplier values. The major theoretical result of the paper is an algorithm which solves the Lagrangian problem in a number of steps proportional to the product ofn ² and the average job processing time. The search for multiplier values which maximize the lower bound leads to the formulation and optimization of the dual problem. The bounds obtained are so sharp that very little enumeration or computer time is required to solve even large problems. Computational experience with 20-, 30-, and 50-job problems is presented.     

On the Continuity of the Optimal Value in Parametric Linear Optimization: Stable Discretization of the Lagrangian Dual of Nonlinear Problems

This paper is focused on the stability of the optimal value, and its immediate repercussion on the stability of the optimal set, for a general parametric family of linear optimization problems in ⁿ. In our approach, the parameter ranges over an arbitrary metric space, and each parameter determines directly a set of coefficient vectors describing the linear system of constraints. Thus, systems associated with different parameters are not required to have the same number (cardinality) of inequalities. In this way, discretization techniques for solving a nominal linear semi-infinite optimization problem may be modeled in terms of suitable parametrized problems. The stability results given in the paper are applied to the stability analysis of the Lagrangian dual associated with a parametric family of nonlinear programming problems. This dual problem is translated into a linear (semi-infinite) programming problem and, then, we prove that the lower semicontinuity of the corresponding feasible set mapping, the continuity of the optimal value function, and the upper semicontinuity of the optimal set mapping are satisfied. Then, the paper shows how these stability properties for the dual problem entail a nice behavior of parametric approximation and discretization strategies (in which an ordinary linear programming problem may be considered in each step). This approximation–discretization process is formalized by means of considering a double parameter: the original one and the finite subset of indices (grid) itself. Finally, the convex case is analyzed, showing that the referred process also allows us to approach the primal problem.Mathematics Subject Classifications (2000) Primary 90C34, 90C31; secondary 90C25, 90C05.     

A proximal point algorithm revisit on the alternating direction method of multipliers

The alternating direction method of multipliers(ADMM)is a benchmark for solving convex programming problems with separable objective functions and linear constraints.In the literature it has been illustrated as an application of the proximal point algorithm(PPA)to the dual problem of the model under consideration.This paper shows that ADMM can also be regarded as an application of PPA to the primal model with a customized choice of the proximal parameter.This primal illustration of ADMM is thus complemental to its dual illustration in the literature.This PPA revisit on ADMM from the primal perspective also enables us to recover the generalized ADMM proposed by Eckstein and Bertsekas easily.A worst-case O(1/t)convergence rate in ergodic sense is established for a slight extension of Eckstein and Bertsekas’s generalized ADMM.     

Solving L 1-CTA in 3D tables by an interior-point method for primal block-angular problems

The purpose of the field of statistical disclosure control is to avoid that confidential information could be derived from statistical data released by, mainly, national statistical agencies. Controlled tabular adjustment (CTA) is an emerging technique for the protection of statistical tabular data. Given a table with sensitive information, CTA looks for the closest safe table. In this work we focus on CTA for three-dimensional tables using the L ₁ norm for the distance between the original and protected tables. Three L ₁-CTA models are presented, giving rise to six different primal block-angular structures of the constraint matrices. The resulting linear programming problems are solved by a specialized interior-point algorithm for this constraints structure which solves the normal equations by a combination of Cholesky factorization and preconditioned conjugate gradients (PCG). In the past this algorithm was shown to be one of the most efficient approaches for some classes of block-angular problems. The effect of quadratic regularizations is also analyzed, showing that for three of the six primal block-angular structures the performance of PCG is guaranteed to improve. Computational results are reported for a set of large instances, which provide linear optimization problems of up to 50 million variables and 25 million constraints. The specialized interior-point algorithm is compared with the state-of-the-art barrier solver of the CPLEX 12.1 package, showing to be a more efficient choice for very large L ₁-CTA instances.     

Updating lower bounds when using Karmarkar's projective algorithm for linear programming

We give two results related to Gonzaga's recent paper showing that lower bounds derived from the Todd-Burrell update can be obtained by solving a one-variable linear programming problem involving the centering direction and the affine direction. We show how these results may be used to update the primal solution when using the dual affine variant of Karmarkar's algorithm. This leads to a dual projective algorithm.This research was partially supported by ONR Grant Number N00014-90-J-1714.     

An exact algorithm for large multiple knapsack problems

The Multiple Knapsack Problem (MKP) is the problem of assigning a subset of n items to m distinct knapsacks, such that the total profit sum of the selected items is maximized, without exceeding the capacity of each of the knapsacks. The problem has several applications in naval as well as financial management. A new exact algorithm for the MKP is presented, which is specially designed for solving large problem instances. The recursive branch-and-bound algorithm applies surrogate relaxation for deriving upper bounds, while lower bounds are obtained by splitting the surrogate solution into the m knapsacks by solving a series of Subset-sum Problems. A new separable dynamic programming algorithm is presented for the solution of Subset-sum Problems, and we also use this algorithm for tightening the capacity constraints in order to obtain better upper bounds. The developed algorithm is compared to the mtm algorithm by Martello and Toth, showing the benefits of the new approach. A surprising result is that large instances with n=100 000 items may be solved in less than a second, and the algorithm has a stable performance even for instances with coefficients in a moderately large range.     

Bound on integrals: Elimination of the dual and reduction of the number of equality constraints

Let L_j (j = 1, …, n + 1) be real linear functions on the convex set $\text{F}$ of probability distributions. We consider the problem of maximization of L_n+1(F) under the constraint F ? $\text{F}$ and the equality constraints L₁(F) = z₁ (i = 1, …, n). Incorporating some of the equality constraints into the basic set $\text{F}$, the problem is equivalent to a problem with less equality constraints. We also show how the dual problems can be eliminated from the statement of the main theorems and we give a new illuminating proof of the existence of particular solutions.The linearity of the functions L_j(j = 1, …, n + 1) can be dropped in several results.