A Projection Method for the Integer Quadratic Knapsack Problem

In this paper we present a new branch and bound algorithm for solving a class of integer quadratic knapsack problems. A previously published algorithm solves the continuous variable subproblems in the branch and bound tree by performing a binary search over the breakpoints of a piecewise linear equation resulting from the Kuhn-Tucker conditions. Here, we first present modifications to a projection method for solving the continuous subproblems. Then we implement the modified projection method in a branch and bound framework and report computational results indicating that the new branch and bound algorithm is superior to the earlier method.     

An SQP-type algorithm for nonlinear second-order cone programs

We propose an SQP-type algorithm for solving nonlinear second-order cone programming (NSOCP) problems. At every iteration, the algorithm solves a convex SOCP subproblem in which the constraints involve linear approximations of the constraint functions in the original problem and the objective function is a convex quadratic function. Those subproblems can be transformed into linear SOCP problems, for which efficient interior point solvers are available. We establish global convergence and local quadratic convergence of the algorithm under appropriate assumptions. We report numerical results to examine the effectiveness of the algorithm. This work was supported in part by the Scientific Research Grant-in-Aid from Japan Society for the Promotion of Science.     

基于D.C.分解的一类箱型约束的非凸二次规划的新型分支定界算法

提出了一类求解带有箱约束的非凸二次规划的新型分支定界算法.首先,把原问题目标函数进行D.C.分解(分解为两个凸函数之差),利用次梯度方法,求出其线性下界逼近函数的一个最优值,也即原问题的一个下界.然后,利用全局椭球算法获得原问题的一个上界,并根据分支定界方法把原问题的求解转化为一系列子问题的求解.最后,理论上证明了算法的收敛性,数值算例表明算法是有效可行的.     

Bounding a class of nonconvex linearly-constrained resource allocation problems via the surrogate dual

We consider a class of problems of resource allocation under economies of scale, namely that of minimizing a lower semicontinuous, isotone, and explicitly quasiconcave cost function subject to linear constraints. An important class of algorithms for the linearly constrained minimization of nonconvex cost functions utilize the branch and bound approach, using convex underestimating cost functions to compute the lower bounds.We suggest instead the use of the surrogate dual problem to bound subproblems. We show that the success of the surrogate dual in fathoming subproblems in a branch and bound algorithm may be determined without directly solving the surrogate dual itself, but that a simple test of the feasibility of a certain linear system of inequalities will suffice. This test is interpreted geometrically and used to characterize the extreme points and extreme rays of the optimal value function's level sets.Research partially supported by NSF under grant # ENG77-06555.     

A Branch and Bound Algorithm for Separable Concave Programming

In this paper, we propose a new branch and bound algorithm for the solution of large scale separable concave programming problems. The largest distance bisection (LDB) technique is proposed to divide rectangle into sub-rectangles when one problem is branched into two subproblems. It is proved that the LDB method is a normal rectangle subdivision(NRS). Numerical tests on problems with dimensions from 100 to 10000 show that the proposed branch and bound algorithm is efficient for solving large scale separable concave programming problems, and convergence rate is faster than ω-subdivision method.     

Budgeting with bounded multiple-choice constraints

We consider a budgeting problem where a specified number of projects from some disjoint classes has to be selected such that the overall gain is largest possible, and such that the costs of the chosen projects do not exceed a fixed upper limit. The problem has several application in government budgeting, planning, and as relaxation from other combinatorial problems. It is demonstrated that the problem can be transformed to an equivalent multiple-choice knapsack problem through dynamic programming. A naive transformation however leads to a drastic increase in the number of variables, thus we propose an algorithm for the continuous problem based on Dantzig–Wolfe decomposition. A master problem solves a continuous multiple-choice knapsack problem knowing only some extreme points in each of the transformed classes. The individual subproblems find extreme points for each given direction, using a median search algorithm. An integer optimal solution is then derived by using the dynamic programming transformation to a multiple-choice knapsack problem for an expanding core. The individual classes are considered in an order given by their gradients, and the transformation to a multiple-choice knapsack problem is performed when needed. In this way, only a dozen of classes need to be transformed for standard instances from the literature. Computational experiments are presented, showing that the developed algorithm is orders of magnitude faster than a general LP/MIP algorithm.     

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.     

Zero-one integer programs with few constraints —Lower bounding theory

The zero-one integer programming problem and its special case, the multiconstraint knapsack problem frequently appear as subproblems in many combinatorial optimization problems. We present several methods for computing lower bounds on the optimal solution of the zero-one integer programming problem. They include Lagrangean, surrogate and composite relaxations. New heuristic procedures are suggested for determining good surrogate multipliers. Based on theoretical results and extensive computational testing, it is shown that for zero-one integer problems with few constraints surrogate relaxation is a viable alternative to the commonly used Lagrangean and linear programming relaxations. These results are used in a follow up paper to develop an efficient branch and bound algorithm for solving zero-one integer programming problems.     

A linear programming-based optimization algorithm for solving nonlinear programming problems

In this paper a linear programming-based optimization algorithm called the Sequential Cutting Plane algorithm is presented. The main features of the algorithm are described, convergence to a Karush–Kuhn–Tucker stationary point is proved and numerical experience on some well-known test sets is showed. The algorithm is based on an earlier version for convex inequality constrained problems, but here the algorithm is extended to general continuously differentiable nonlinear programming problems containing both nonlinear inequality and equality constraints. A comparison with some existing solvers shows that the algorithm is competitive with these solvers. Thus, this new method based on solving linear programming subproblems is a good alternative method for solving nonlinear programming problems efficiently. The algorithm has been used as a subsolver in a mixed integer nonlinear programming algorithm where the linear problems provide lower bounds on the optimal solutions of the nonlinear programming subproblems in the branch and bound tree for convex, inequality constrained problems.     

Lower Bound Improvement and Forcing Rule for Quadratic Binary Programming

In this paper several equivalent formulations for the quadratic binary programming problem are presented. Based on these formulations we describe four different kinds of strategies for estimating lower bounds of the objective function, which can be integrated into a branch and bound algorithm for solving the quadratic binary programming problem. We also give a theoretical explanation for forcing rules used to branch the variables efficiently, and explore several properties related to obtained subproblems. From the viewpoint of the number of subproblems solved, new strategies for estimating lower bounds are better than those used before. A variant of a depth-first branch and bound algorithm is described and its numerical performance is presented.     

Iterative estimation maximization for stochastic linear programs with conditional value-at-risk constraints

We present a new algorithm, iterative estimation maximization (IEM), for stochastic linear programs with conditional value-at-risk constraints. IEM iteratively constructs a sequence of linear optimization problems, and solves them sequentially to find the optimal solution. The size of the problem that IEM solves in each iteration is unaffected by the size of random sample points, which makes it extremely efficient for real-world, large-scale problems. We prove the convergence of IEM, and give a lower bound on the number of sample points required to probabilistically bound the solution error. We also present computational performance on large problem instances and a financial portfolio optimization example using an S&P 500 data set.     

A column generation based decomposition algorithm for a parallel machine just-in-time scheduling problem

We propose a column generation based exact decomposition algorithm for the problem of scheduling n jobs with an unrestrictively large common due date on m identical parallel machines to minimize total weighted earliness and tardiness. We first formulate the problem as an integer program, then reformulate it, using Dantzig–Wolfe decomposition, as a set partitioning problem with side constraints. Based on this set partitioning formulation, a branch and bound exact solution algorithm is developed for the problem. In the branch and bound tree, each node is the linear relaxation problem of a set partitioning problem with side constraints. This linear relaxation problem is solved by column generation approach where columns represent partial schedules on single machines and are generated by solving two single machine subproblems. Our computational results show that this decomposition algorithm is capable of solving problems with up to 60 jobs in reasonable cpu time.     

A simplicial branch and duality bound algorithm for the sum of convex–convex ratios problem

This article presents a simplicial branch and duality bound algorithm for globally solving the sum of convex–convex ratios problem with nonconvex feasible region. To our knowledge, little progress has been made for globally solving this problem so far. The algorithm uses a branch and bound scheme where the Lagrange duality theory is used to obtain the lower bounds. As a result, the lower-bounding subproblems during the algorithm search are all ordinary linear programs that can be solved very efficiently. It has been proved that the algorithm possesses global convergence. Finally, the numerical experiments are given to show the feasibility of the proposed algorithm.     

A branch and bound algorithm with constraint partitioning for integer goal programming problems

Using constraint partitioning and variable elimination, the authors have recently developed an efficient algorithm for solving linear goal programming problems. However, many goal programs require some or all of the decision variables to be integer valued. This paper shows how the new partitioning algorithm can be extended with a modified branch and bound strategy to solve both pure and mixed type integer goal programming problems. A potential problem in combining the partitioning algorithm and the branch and bound search scheme is presented and resolved.     

Inference of statistical bounds for multistage stochastic programming problems

We discuss in this paper statistical inference of sample average approximations of multistage stochastic programming problems. We show that any random sampling scheme provides a valid statistical lower bound for the optimal (minimum) value of the true problem. However, in order for such lower bound to be consistent one needs to employ the conditional sampling procedure. We also indicate that fixing a feasible first-stage solution and then solving the sampling approximation of the corresponding (T–1)-stage problem, does not give a valid statistical upper bound for the optimal value of the true problem.Supported, in part, by the National Science Foundation under grant DMS-0073770.     

An Adaptive Dual Parametrization Algorithm for Quadratic Semi-infinite Programming Problems

The so called dual parametrization method for quadratic semi-infinite programming (SIP) problems is developed recently for quadratic SIP problems with a single infinite constraint. A dual parametrization algorithm is also proposed for numerical solution of such problems. In this paper, we consider quadratic SIP problems with positive definite objective and multiple linear infinite constraints. All the infinite constraints are supposed to be continuously dependent on their index variable on a compact set which is defined by a number equality and inequalities. We prove that in the multiple infinite constraint case, the minimu parametrization number, just as in the single infinite constraint case, is less or equal to the dimension of the SIP problem. Furthermore, we propose an adaptive dual parametrization algorithm with convergence result. Compared with the previous dual parametrization algorithm, the adaptive algorithm solves subproblems with much smaller number of constraints. The efficiency of the new algorithm is shown by solving a number of numerical examples.     

A general algorithm for solving Generalized Geometric Programming with nonpositive degree of difficulty

In this paper, a general algorithm for solving Generalized Geometric Programming with nonpositive degree of difficulty is proposed. It shows that under certain assumptions the primal problem can be transformed and decomposed into several subproblems which are easy to solve, and furthermore we verify that through solving these subproblems we can obtain the optimal value and solutions of the primal problem which are global solutions. At last, some examples are given to vindicate our conclusions.     

A successive quadratic programming method for a class of constrained nonsmooth optimization problems

In this paper we present an algorithm for solving nonlinear programming problems where the objective function contains a possibly nonsmooth convex term. The algorithm successively solves direction finding subproblems which are quadratic programming problems constructed by exploiting the special feature of the objective function. An exact penalty function is used to determine a step-size, once a search direction thus obtained is judged to yield a sufficient reduction in the penalty function value. The penalty parameter is adjusted to a suitable value automatically. Under appropriate assumptions, the algorithm is shown to produce an approximate optimal solution to the problem with any desirable accuracy in a finite number of iterations.     

Using concave envelopes to globally solve the nonlinear sum of ratios problem

This article presents a branch and bound algorithm for globally solving the nonlinear sum of ratios problem (P). The algorithm works by globally solving a sum of ratios problem that is equivalent to problem (P). In the algorithm, upper bounds are computed by maximizing concave envelopes of a sum of ratios function over intersections of the feasible region of the equivalent problem with rectangular sets. The rectangular sets are systematically subdivided as the branch and bound search proceeds. Two versions of the algorithm, with convergence results, are presented. Computational advantages of these algorithms are indicated, and some computational results are given that were obtained by globally solving some sample problems with one of these algorithms.     

An arc-exchange decomposition method for multistage dynamic networks with random arc capacities

Multistage dynamic networks with random arc capacities (MDNRAC) have been successfully used for modeling various resource allocation problems in the transportation area. However, solving these problems is generally computationally intensive, and there is still a need to develop more efficient solution approaches. In this paper, we propose a new heuristic approach that solves the MDNRAC problem by decomposing the network at each stage into a series of subproblems with tree structures. Each subproblem can be solved efficiently. The main advantage is that this approach provides an efficient computational device to handle the large-scale problem instances with fairly good solution quality. We show that the objective value obtained from this decomposition approach is an upper bound for that of the MDNRAC problem. Numerical results demonstrate that our proposed approach works very well.