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.     

Lagrangean/surrogate relaxation for generalized assignment problems

This work presents Lagrangean/surrogate relaxation to the problem of maximum profit assignment of n tasks to m agents (n > m), such that each task is assigned to only one agent subject to capacity constraints on the agents. The Lagrangean/surrogate relaxation combines usual Lagrangean and surrogate relaxations relaxing first a set of constraints in the surrogate way. Then, the Lagrangean relaxation of the surrogate constraint is obtained and approximately optimized (one-dimensional dual). The Lagrangean/surrogate is compared with the usual Lagrangean relaxation on a computational study using a large set of instances. The dual bounds are the same for both relaxations, but the Lagrangean/surrogate can give improved local bounds at the application of a subgradient method, resulting in less computational times. Three relaxations are derived for the problem. The first relaxation considers a vector of multipliers for the capacity constraints, the second for the assignment constraints and the other for the Lagrangean decomposition constraints. Relaxation multipliers are used with efficient constructive heuristics to find good feasible solutions. The application of a Lagrangean/surrogate approach seems very promising for large scale problems.     

Upper and lower bounding procedures for the multiple knapsack assignment problem

We formulate the multiple knapsack assignment problem (MKAP) as an extension of the multiple knapsack problem (MKP), as well as of the assignment problem. Except for small instances, MKAP is hard to solve to optimality. We present a heuristic algorithm to solve this problem approximately but very quickly. We first discuss three approaches to evaluate its upper bound, and prove that these methods compute an identical upper bound. In this process, reference capacities are derived, which enables us to decompose the problem into mutually independent MKPs. These MKPs are solved euristically, and in total give an approximate solution to MKAP. Through numerical experiments, we evaluate the performance of our algorithm. Although the algorithm is weak for small instances, we find it prospective for large instances. Indeed, for instances with more than a few thousand items we usually obtain solutions with relative errors less than 0.1% within one CPU second.     

Entropic perturbation method for solving a system of linear inequalities

The problem of finding an $\text{x∈}\text{R}{}^{\mathrm{n}}$ such that Ax⩽b and x⩾0 arises in numerous contexts. We propose a new optimization method for solving this feasibility problem. After converting Ax⩽b into a system of equations by introducing a slack variable for each of the linear inequalities, the method imposes an entropy function over both the original and the slack variables as the objective function. The resulting entropy optimization problem is convex and has an unconstrained convex dual. If the system is consistent and has an interior solution, then a closed-form formula converts the dual optimal solution to the primal optimal solution, which is a feasible solution for the original system of linear inequalities. An algorithm based on the Newton method is proposed for solving the unconstrained dual problem. The proposed algorithm enjoys the global convergence property with a quadratic rate of local convergence. However, if the system is inconsistent, the unconstrained dual is shown to be unbounded. Moreover, the same algorithm can detect possible inconsistency of the system. Our numerical examples reveal the insensitivity of the number of iterations to both the size of the problem and the distance between the initial solution and the feasible region. The performance of the proposed algorithm is compared to that of the surrogate constraint algorithm recently developed by Yang and Murty. Our comparison indicates that the proposed method is particularly suitable when the number of constraints is larger than that of the variables and the initial solution is not close to the feasible region.     

Minimax linear programming problem

In this paper, we consider the following minimax linear programming problem: min z = max_{1 ≤ j ≤ n}{C_jX_j}, subject to Ax = g, x ≥ 0. It is well known that this problem can be transformed into a linear program by introducing n additional constraints. We note that these additional constraints can be considered implicitly by treating them as parametric upper bounds. Based on this approach we develop two algorithms: a parametric algorithm and a primal—dual algorithm. The parametric algorithm solves a linear programming problem with parametric upper bounds and the primal—dual algorithm solves a sequence of related dual feasible linear programming problems. Computation results are also presented, which indicate that both the algorithms are substantially faster than the simplex algorithm applied to the enlarged linear programming problem.     

Heuristics for the 0–1 multidimensional knapsack problem

Two heuristics for the 0–1 multidimensional knapsack problem (MKP) are presented. The first one uses surrogate relaxation, and the relaxed problem is solved via a modified dynamic-programming algorithm. The heuristics provides a feasible solution for (MKP). The second one combines a limited-branch-and-cut-procedure with the previous approach, and tries to improve the bound obtained by exploring some nodes that have been rejected by the modified dynamic-programming algorithm. Computational experiences show that our approaches give better results than the existing heuristics, and thus permit one to obtain a smaller gap between the solution provided and an optimal solution.     

A general theory of surrogate dual and perturbational extended surrogate dual optimization problems

For the optimization problem (P) α = inf h(G), where G ≠ Ø is a subset of a locally convex space $\text{F}\text{and}\text{h: F →}\text{R}\text{?}$, we introduce and study two general concepts of dual problems, encompassing the classical surrogate dual problem. The first one involves only a family of surrogate constraints sets Δ_{G, Φ} ? F (Φ ∈ W), where W ? R^X, X being a locally convex space. The second one uses a perturbation functional $\text{?: F × X →}\text{R}\text{?}$ and a family of sets $\text{}\text{?}\text{gD}{}_{\mathrm{(F,x0),\Phi }}\text{? F × X (Φ ∈ W)}$, where W ? R^X. We give duality theorems, introduce Lagrangians, and show some relations between these problems and the dual problems to (P) defined with the aid of a perturbation and a concept of conjugation of functionals.     

A Lexicographic 0.5-Approximation Algorithm for the Multiple Knapsack Problem

We present a 0.5-approximation algorithm for the Multiple Knapsack Problem (MKP). The algorithm uses the ordering of knapsacks according to the nondecreasing of size and the two orderings of items: in nonincreasing utility order and in nonincreasing order of the utility/size ratio. These orderings create two lexicographic orderings on A × B (here A is the set of knapsacks and B is the set of indivisible items). Based on each of these lexicographic orderings, the algorithm creates a feasible solution to the MKP by looking through the pairs (a, b) ∈ A × B in the corresponding order and placing item b into knapsack a if this item is not placed yet and there is enough free space in the knapsack. The algorithm chooses the best of the two obtained solutions. This algorithm is 0.5-approximate and has runtime O(mn) (without sorting), where mand n are the sizes of A and B correspondingly.     

Equivalence of communication and projective boundedness properties for monotone and homogeneous functions

This work concerns the eigenvalue problem for a monotone and homogeneous self-mapping f of a finite-dimensional positive cone. A communication criterion is formulated such that it is equivalent to the projective boundedness of the upper eigenspaces associated with f, a property that yields the existence of a nonlinear eigenvalue. Using the idea of dual function, a similar result is obtained for lower eigenspaces.     

A branch-and-bound algorithm for hard multiple knapsack problems

The multiple knapsack problem (MKP) is a classical combinatorial optimization problem. A recent algorithm for some classes of the MKP is bin-completion, a bin-oriented, branch-and-bound algorithm. In this paper, we propose path-symmetry and path-dominance criteria for pruning nodes in the MKP branch-and-bound search space. In addition, we integrate the ??bound-and-bound?? upper bound validation technique used in previous MKP solvers. We show experimentally that our new MKP solver, which successfully integrates dominance based pruning, symmetry breaking, and bound-and-bound, significantly outperforms previous solvers on some classes of hard problem instances.     

Dual bivariational principles for linear problems

This paper presents dual bivariational principles which yield upper and lower bounds for 〈g, φ〉, where g is an arbitrary function and φ is the solution of the linear equation Aφ = f with general mixed boundary conditions. Variational principles associated with 〈f, φ〉 are taken as the starting-point, and the results generalize those of recent authors for linear integral equations.     

Entropic Value-at-Risk: A New Coherent Risk Measure

This paper introduces the concept of entropic value-at-risk (EVaR), a new coherent risk measure that corresponds to the tightest possible upper bound obtained from the Chernoff inequality for the value-at-risk (VaR) as well as the conditional value-at-risk (CVaR). We show that a broad class of stochastic optimization problems that are computationally intractable with the CVaR is efficiently solvable when the EVaR is incorporated. We also prove that if two distributions have the same EVaR at all confidence levels, then they are identical at all points. The dual representation of the EVaR is closely related to the Kullback-Leibler divergence, also known as the relative entropy. Inspired by this dual representation, we define a large class of coherent risk measures, called g-entropic risk measures. The new class includes both the CVaR and the EVaR.     

On the continuity of the maximum in parametric linear programming

For the widest class of parametric linear programs with continuous dependence of coefficients on parameters, the following theorem is proven: for any parameter vectort ₀ in the domain of definition of the maximum, if the set of optimal solutions is bounded, then the maximum is upper semicontinuous att ₀. If the same proviso is met also in the dual program, then the maximum must be continuous att ₀.     

On lower and upper bounds of matrices

Using an approach of Bergh, we give an alternate proof of Bennett's result on lower bounds for non-negative matrices acting on non-increasing non-negative sequences in lp when p?1 and its dual version, the upper bounds when 0<p?1. We also determine such bounds explicitly for some families of matrices.     

pth Power Lagrangian Method for Integer Programming

When does there exist an optimal generating Lagrangian multiplier vector (that generates an optimal solution of an integer programming problem in a Lagrangian relaxation formulation), and in cases of nonexistence, can we produce the existence in some other equivalent representation space? Under what conditions does there exist an optimal primal-dual pair in integer programming? This paper considers both questions. A theoretical characterization of the perturbation function in integer programming yields a new insight on the existence of an optimal generating Lagrangian multiplier vector, the existence of an optimal primal-dual pair, and the duality gap. The proposed pth power Lagrangian method convexifies the perturbation function and guarantees the existence of an optimal generating Lagrangian multiplier vector. A condition for the existence of an optimal primal-dual pair is given for the Lagrangian relaxation method to be successful in identifying an optimal solution of the primal problem via the maximization of the Lagrangian dual. The existence of an optimal primal-dual pair is assured for cases with a single Lagrangian constraint, while adopting the pth power Lagrangian method. This paper then shows that an integer programming problem with multiple constraints can be always converted into an equivalent form with a single surrogate constraint. Therefore, success of a dual search is guaranteed for a general class of finite integer programming problems with a prominent feature of a one-dimensional dual search.     

Advanced greedy algorithms and surrogate constraint methods for linear and quadratic knapsack and covering problems

New variants of greedy algorithms, called advanced greedy algorithms, are identified for knapsack and covering problems with linear and quadratic objective functions. Beginning with single-constraint problems, we provide extensions for multiple knapsack and covering problems, in which objects must be allocated to different knapsacks and covers, and also for multi-constraint (multi-dimensional) knapsack and covering problems, in which the constraints are exploited by means of surrogate constraint strategies. In addition, we provide a new graduated-probe strategy for improving the selection of variables to be assigned values. Going beyond the greedy and advanced greedy frameworks, we describe ways to utilize these algorithms with multi-start and strategic oscillation metaheuristics. Finally, we identify how surrogate constraints can be utilized to produce inequalities that dominate those previously proposed and tested utilizing linear programming methods for solving multi-constraint knapsack problems, which are responsible for the current best methods for these problems. While we focus on 0–1 problems, our approaches can readily be adapted to handle variables with general upper bounds.     

Near convexity,near smoothness and approximative compactness of half spaces in Banach spaces

The authors discuss the dual relation of nearly very convexity and property WS. By two kinds of near convexities and two kinds of near smoothness, the authors prove a series of characterizations such that every half-space in Banach space X and every weak* half-space in the dual space X* are approximatively weakly compact and approximatively compact. They show a sufficient condition such that a Banach space X is a Asplund space. Using upper semi-continuity of duality mapping, the authors also give two characterizations of property WS and property S.     

Job-shop local-search move evaluation without direct consideration of the criterion’s value

This article focuses on the evaluation of moves for the local search of the job-shop problem with the makespan criterion. We reason that the omnipresent ranking of moves according to their resulting value of a criterion function makes the local search unnecessarily myopic. Consequently, we introduce an alternative evaluation that relies on a surrogate quantity of the move’s potential, which is related to, but not strongly coupled with, the bare criterion. The approach is confirmed by empirical tests, where the proposed evaluator delivers a new upper bound on the well-known benchmark test yn2. The line of the argumentation also shows that by sacrificing accuracy the established makespan estimators unintentionally improve on the move evaluation in comparison to the exact makespan calculation, in contrast to the belief that the reliance on estimation degrades the optimization results.