Lower and upper bounding procedures for the bin packing problem with concave loading cost

We address the one-dimensional bin packing problem with concave loading cost (BPPC), which commonly arises in less-than-truckload shipping services. Our contribution is twofold. First, we propose three lower bounds for this problem. The first one is the optimal solution of the continuous relaxation of the problem for which a closed form is proposed. The second one allows the splitting of items but not the fractioning of bins. The third one is based on a large-scale set partitioning formulation of the problem. In order to circumvent the challenges posed by the non-linearity of the objective function coefficients, we considered the inner-approximation of the concave load cost and derived a relaxed formulation that is solved by column generation. In addition, we propose two subset-sum-based heuristics. The first one is a constructive heuristic while the second one is a local search heuristic that iteratively attempts to improve the current solution by selecting pairs of bins and solving the corresponding subset sum-problem. We show that the worst-case performance of any BPPC heuristic and any concave loading cost function is bounded by 2. We present the results of an extensive computational study that was carried out on large set of benchmark instances. This study provides empirical evidence that the column generation-based lower bound and the local search heuristic consistently exhibit remarkable performance.     

A computational comparison of some branch and bound methods for indefinite quadratic programs

The aim of this paper is to discuss different branch and bound methods for solving indefinite quadratic programs. In these methods the quadratic objective function is decomposed in a d.c. form and the relaxations are obtained by linearizing the concave part of the decomposition. In this light, various decomposition schemes have been considered and studied. The various branch and bound solution methods have been implemented and compared by means of a deep computational test.      

Exact Algorithm for Concave Knapsack Problems: Linear Underestimation and Partition Method

Integer programming problems with a concave cost function are often encountered in optimization models involving economics of scale. In this paper, we propose an efficient exact algorithm for solving concave knapsack problems. The algorithm consists of an iterative process between finding lower and upper bounds by linearly underestimating the objective function and performing domain cut and partition by exploring the special structure of the problem. The lower bound is improved iteratively via cutting and partitioning the domain. This iteration process converges to the optimality in a finite number of steps. Promising computational results are reported for large-scale concave knapsack problems with up to 1200 integer variables. Comparison results with other existing methods in the literature are also presented. *Research supported by the National Natural Science Foundation of China under Grants 79970107 and 10271073,and the Research Grants Council of Hong Kong under Grant CUHK 4214/01E.     

Pricing Bounds for Volatility Derivatives via Duality and Least Squares Monte Carlo

Derivatives on the Chicago Board Options Exchange volatility index have gained significant popularity over the last decade. The pricing of volatility derivatives involves evaluating the square root of a conditional expectation which cannot be computed by direct Monte Carlo methods. Least squares Monte Carlo methods can be used, but the sign of the error is difficult to determine. In this paper, we propose a new model-independent technique for computing upper and lower pricing bounds for volatility derivatives. In particular, we first present a general stochastic duality result on payoffs involving convex (or concave) functions. This result also allows us to interpret these contingent claims as a type of chooser options. It is then applied to volatility derivatives along with minor adjustments to handle issues caused by the square root function. The upper bound involves the evaluation of a variance swap, while the lower bound involves estimating a martingale increment corresponding to its hedging portfolio. Both can be achieved simultaneously using a single linear least square regression. Numerical results show that the method works very well for futures, calls and puts under a wide range of parameter choices.     

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.     

Improve-and-Branch Algorithm for the Global Optimization of Nonconvex NLP Problems

A new algorithm to solve nonconvex NLP problems is presented. It is based on the solution of two problems. The reformulated problem RP is a suitable reformulation of the original problem and involves convex terms and concave univariate terms. The main problem MP is a nonconvex NLP that outer-approximates the feasible region and underestimate the objective function. MP involves convex terms and terms which are the products of concave univariate functions and new variables. Fixing the variables in the concave terms, a convex NLP that overestimates the feasible region and underestimates the objective function is obtained from the MP. Like most of the deterministic global optimization algorithms, bounds on all the variables in the nonconvex terms must be provided. MP forces the objective value to improve and minimizes the difference of upper and lower bound of all the variables either to zero or to a positive value. In the first case, a feasible solution of the original problem is reached and the objective function is improved. In general terms, the second case corresponds to an infeasible solution of the original problem due to the existence of gaps in some variables. A branching procedure is performed in order to either prove that there is no better solution or reduce the domain, eliminating the local solution of MP that was found. The MP solution indicates a key point to do the branching. A bound reduction technique is implemented to accelerate the convergence speed. Computational results demonstrate that the algorithm compares very favorably to other approaches when applied to test problems and process design problems. It is typically faster and it produces very accurate results.     

Global optimization of concave functions subject to quadratic constraints: An application in nonlinear bilevel programming

When the follower's optimality conditions are both necessary and sufficient, the nonlinear bilevel program can be solved as a global optimization problem. The complementary slackness condition is usually the complicating constraint in such problems. We show how this constraint can be replaced by an equivalent system of convex and separable quadratic constraints. In this paper, we propose different methods for finding the global minimum of a concave function subject to quadratic separable constraints. The first method is of the branch and bound type, and is based on rectangular partitions to obtain upper and lower bounds. Convergence of the proposed algorithm is also proved. For computational purposes, different procedures that accelerate the convergence of the proposed algorithm are analysed. The second method is based on piecewise linear approximations of the constraint functions. When the constraints are convex, the problem is reduced to global concave minimization subject to linear constraints. In the case of non-convex constraints, we use zero-one integer variables to linearize the constraints. The number of integer variables depends only on the concave parts of the constraint functions.Parts of the present paper were prepared while the second author was visiting Georgia Tech and the University of Florida.     

A parallel algorithm for constrained concave quadratic global minimization

The global minimization of large-scale concave quadratic problems over a bounded polyhedral set using a parallel branch and bound approach is considered. The objective function consists of both a concave part (nonlinear variables) and a strictly linear part, which are coupled by the linear constraints. These large-scale problems are characterized by having the number of linear variables much greater than the number of nonlinear variables. A linear underestimating function to the concave part of the objective is easily constructed and minimized over the feasible domain to get both upper and lower bounds on the global minimum function value. At each minor iteration of the algorithm, the feasible domain is divided into subregions and linear underestimating problems over each subregion are solved in parallel. Branch and bound techniques can then be used to eliminate parts of the feasible domain from consideration and improve the upper and lower bounds. It is shown that the algorithm guarantees that a solution is obtained to within any specified tolerance in a finite number of steps. Computational results are presented for problems with 25 and 50 nonlinear variables and up to 400 linear variables. These results were obtained on a four processor CRAY2 using both sequential and parallel implementations of the algorithm. The average parallel solution time was approximately 15 seconds for problems with 400 linear variables and a relative tolerance of 0.001. For a relative tolerance of 0.1, the average computation time appears to increase only linearly with the number of linear variables.     

Machine assignment in a nonlinear multi-product flowline

A group of machines for processing a set of jobs in a manufacturing system is often located in a serial line. An efficient strategy for locating these machines such that the total travel distance or the cost of transporting the jobs is minimized is desired. In this research, the assumption of a linear line with equally spaced machine location is relaxed. This research addressed problems of locating unique machines. It is found that the machine distances possess unique properties in this type of a problem. Utilizing these properties, heuristic strategies are proposed to obtain efficient solution where optimal methods are expected to be computationally prohibitive. A lower bound for the optimum solution is also proposed. Results are encouraging.     

Valid inequalities for concave piecewise linear regression

We consider the problem of fitting a concave piecewise linear function to multivariate data using the Least Absolute Deviation objective. We propose new valid inequalities for the problem using the properties of concave functions. Results with univariate data show that the proposed valid inequalities improve the root relaxation lower bound, permitting significant improvements in solution time.     

A Lower Bound for the Penalty Parameter in the Exact Minimax Penalty Function Method for Solving Nondifferentiable Extremum Problems

In the paper, we consider the exact minimax penalty function method used for solving a general nondifferentiable extremum problem with both inequality and equality constraints. We analyze the relationship between an optimal solution in the given constrained extremum problem and a minimizer in its associated penalized optimization problem with the exact minimax penalty function under the assumption of convexity of the functions constituting the considered optimization problem (with the exception of those equality constraint functions for which the associated Lagrange multipliers are negative—these functions should be assumed to be concave). The lower bound of the penalty parameter is given such that, for every value of the penalty parameter above the threshold, the equivalence holds between the set of optimal solutions in the given extremum problem and the set of minimizers in its associated penalized optimization problem with the exact minimax penalty function.     

On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems,including inequalities for log concave functions,and with an application to the diffusion equation

We extend the Prékopa-Leindler theorem to other types of convex combinations of two positive functions and we strengthen the Prékopa-Leindler and Brunn-Minkowski theorems by introducing the notion of essential addition. Our proof of the Prékopa-Leindler theorem is simpler than the original one. We sharpen the inequality that the marginal of a log concave function is log concave, and we prove various moment inequalities for such functions. Finally, we use these results to derive inequalities for the fundamental solution of the diffusion equation with a convex potential.     

Finsler流形上的Laplace比较定理和有界估计(英文)

本文研究了Finsler流形上的距离函数的Laplacian.利用指标引理和文献[4]中主要方法,获得了Ricci曲率有函数下界的Laplacian比较定理,改进了文献[6]和文献[7]的相关结果.     

Lower Bounds for Measurable Chromatic Numbers

The Lovász theta function provides a lower bound for the chromatic number of finite graphs based on the solution of a semidefinite program. In this paper we generalize it so that it gives a lower bound for the measurable chromatic number of distance graphs on compact metric spaces. In particular we consider distance graphs on the unit sphere. There we transform the original infinite semidefinite program into an infinite linear program which then turns out to be an extremal question about Jacobi polynomials which we solve explicitly in the limit. As an application we derive new lower bounds for the measurable chromatic number of the Euclidean space in dimensions 10, . . . , 24 and we give a new proof that it grows exponentially with the dimension.     

An improvement of level set equations via approximation of a distance function

In the classical level set method, the slope of solutions can be very small or large, and it can make it difficult to get the precise level set numerically. In this paper, we introduce an improved level set equation whose solutions are close to the signed distance function to evolving interfaces. The improved equation is derived via approximation of the evolution equation for the distance function. Applying the comparison principle, we give an upper- and lower bound near the zero level set for the viscosity solution to the initial value problem.     

A parallel algorithm for partially separable non-convex global minimization: Linear constraints

The global minimization of large-scale partially separable non-convex problems over a bounded polyhedral set using a parallel branch and bound approach is considered. The objective function consists of a separable concave part, an unseparated convex part, and a strictly linear part, which are all coupled by the linear constraints. These large-scale problems are characterized by having the number of linear variables much greater than the number of nonlinear variables. An important special class of problems which can be reduced to this form are the synomial global minimization problems. Such problems often arise in engineering design, and previous computational methods for such problems have been limited to the convex posynomial case. In the current work, a convex underestimating function to the objective function is easily constructed and minimized over the feasible domain to get both upper and lower bounds on the global minimum function value. At each minor iteration of the algorithm, the feasible domain is divided into subregions and convex underestimating problems over each subregion are solved in parallel. Branch and bound techniques can then be used to eliminate parts of the feasible domain from consideration and improve the upper and lower bounds. It is shown that the algorithm guarantees that a solution is obtained to within any specified tolerance in a finite number of steps. Computational results obtained on the four processor Cray 2, both sequentially and in parallel on all four processors, are also presented.     

Conformal metrics with prescribed Q-curvature on S n

In this paper we prescribe a fourth order conformal invariant on the standard n-sphere, with n????5, and study the related fourth order elliptic equation. We prove new existence results based on a new type of Euler?CHopf type formula. Our argument gives an upper bound on the Morse index of the obtained solution. We also give a lower bound on the number of conformal metrics having the same Q-curvature.     

A finite concave minimization algorithm using branch and bound and neighbor generation

In this article we present a new finite algorithm for globally minimizing a concave function over a compact polyhedron. The algorithm combines a branch and bound search with a new process called neighbor generation. It is guaranteed to find an exact, extreme point optimal solution, does not require the objective function to be separable or even analytically defined, requires no nonlinear computations, and requires no determinations of convex envelopes or underestimating functions. Linear programs are solved in the branch and bound search which do not grow in size and differ from one another in only one column of data. Some preliminary computational experience is also presented.     

A note on adapting methods for continuous global optimization to the discrete case

In this note we show that various branch and bound methods for solving continuous global optimization problems can be readily adapted to the discrete case. As an illustration, we present an algorithm for minimizing a concave function over the integers contained in a compact polyhedron. Computational experience with this algorithm is reported.