A mathematical programming model and solution for scheduling production orders in Shanghai Baoshan Iron and Steel Complex

In this paper, we investigate the production order scheduling problem derived from the production of steel sheets in Shanghai Baoshan Iron and Steel Complex (Baosteel). A deterministic mixed integer programming (MIP) model for scheduling production orders on some critical and bottleneck operations in Baosteel is presented in which practical technological constraints have been considered. The objective is to determine the starting and ending times of production orders on corresponding operations under capacity constraints for minimizing the sum of weighted completion times of all orders. Due to large numbers of variables and constraints in the model, a decomposition solution methodology based on a synergistic combination of Lagrangian relaxation, linear programming and heuristics is developed. Unlike the commonly used method of relaxing capacity constraints, this methodology alternatively relaxes constraints coupling integer variables with continuous variables which are introduced to the objective function by Lagrangian multipliers. The Lagrangian relaxed problem can be decomposed into two sub-problems by separating continuous variables from integer ones. The sub-problem that relates to continuous variables is a linear programming problem which can be solved using standard software package OSL, while the other sub-problem is an integer programming problem which can be solved optimally by further decomposition. The subgradient optimization method is used to update Lagrangian multipliers. A production order scheduling simulation system for Baosteel is developed by embedding the above Lagrangian heuristics. Computational results for problems with up to 100 orders show that the proposed Lagrangian relaxation method is stable and can find good solutions within a reasonable time.     

Nonnormal deterministic equivalents and a transformation in stochastic mathematical programming

This paper considers random variables of the continuous type in a stochastic programming problem and presents (1) a general approach to the development of deterministic equivalents of constraints to be satisfied within certain probability limits, and (2) a deterministic transformation of a stochastic programming problem with random variables in the objective function. Deterministic equivalents are developed for constraints containing uniform random variables, but the approach used can be applied to other types of continuous random variables, as well. When the random variables appear in the objective function, a deterministic transformation of the stochastic programming problem is obtained to yield a closed-form solution without resort to a Monte Carlo computer simulation. Extension of this approach to stochastic problems with discrete random variables and integer decision variables is discussed briefly. A numerical example is presented.     

Disjunctive cuts for continuous linear bilevel programming

This work shows how disjunctive cuts can be generated for a bilevel linear programming problem (BLP) with continuous variables. First, a brief summary on disjunctive programming and bilevel programming is presented. Then duality theory is used to reformulate BLP as a disjunctive program and, from there, disjunctive programming results are applied to derive valid cuts. These cuts tighten the domain of the linear relaxation of BLP. An example is given to illustrate this idea, and a discussion follows on how these cuts may be incorporated in an algorithm for solving BLP.     

A general MPCC model and its solution algorithm for continuous network design problem

This paper formulates the continuous network design problem as a mathematical program with complementarity constraints (MPCC), with the upper level a nonlinear programming problem and the lower level a nonlinear complementarity problem. Unlike in most previous studies, the proposed framework is more general, in which both symmetric and asymmetric user equilibria can be captured. By applying the complementarity slackness condition of the lower-level problem, the original bilevel formulation can be converted into a single-level and smooth nonlinear programming problem. In order to solve the problem, a relaxation scheme is applied by progressively restricting the complementarity condition, which has been proven to be a rigorous approach under certain conditions. The model and solution algorithm are tested for well-known network design problems and promising results are shown.     

基于正弦型光滑打磨函数对0-1规划问题的连续化求解方法

传统的求解0-1规划问题方法大多属于直接离散的解法.现提出一个包含严格转换和近似逼近三个步骤的连续化解法:(1)借助阶跃函数把0-1离散变量转化为[0,1]区间上的连续变量;(2)对目标函数采用逼近折中阶跃函数近光滑打磨函数,约束条件采用线性打磨函数逼近折中阶跃函数,把0-1规划问题由离散问题转化为连续优化模型;(3)利用高阶光滑的解法求解优化模型.该方法打破了特定求解方法仅适用于特定类型0-1规划问题惯例,使求解0-1规划问题的方法更加一般化.在具体求解时,采用正弦型光滑打磨函数来逼近折中阶跃函数,计算效果很好.     

Design of induction motors using a mixed-variable approach

In this paper we are concerned with the problem of optimally designing three-phase induction motors. This problem can be formulated as a mixed variable programming problem. Two different solution strategies have been used to solve this problem. The first one consists in solving the continuous nonlinear optimization problem obtained by suitably relaxing the discrete variables. On the opposite, the second strategy tries to manage directly the discrete variables by alternating a continuous search phase and a discrete search phase. The comparison between the numerical results obtained with the above two strategies clearly shows the fruitfulness of taking directly into account the presence of both continuous and discrete variables.This work was supported by CNR/MIUR Research Program “Metodi e sistemi di supporto alle decisioni”, Rome, Italy.     

A polyhedral study of the cardinality constrained knapsack problem

 A cardinality constrained knapsack problem is a continuous knapsack problem in which no more than a specified number of nonnegative variables are allowed to be positive. This structure occurs, for example, in areas such as finance, location, and scheduling. Traditionally, cardinality constraints are modeled by introducing auxiliary 0-1 variables and additional constraints that relate the continuous and the 0-1 variables. We use an alternative approach, in which we keep in the model only the continuous variables, and we enforce the cardinality constraint through a specialized branching scheme and the use of strong inequalities valid for the convex hull of the feasible set in the space of the continuous variables. To derive the valid inequalities, we extend the concepts of cover and cover inequality, commonly used in 0-1 programming, to this class of problems, and we show how cover inequalities can be lifted to derive facet-defining inequalities. We present three families of non-trivial facet-defining inequalities that are lifted cover inequalities. Finally, we report computational results that demonstrate the effectiveness of lifted cover inequalities and the superiority of the approach of not introducing auxiliary 0-1 variables over the traditional MIP approach for this class of problems. Received: March 13, 2003 Published online: April 10, 2003 Key Words. mixed-integer programming – knapsack problem – cardinality constrained programming – branch-and-cut     

Development of a new approach for deterministic supply chain network design

This paper proposes a mixed integer linear programming model and solution algorithm for solving supply chain network design problems in deterministic, multi-commodity, single-period contexts. The strategic level of supply chain planning and tactical level planning of supply chain are aggregated to propose an integrated model. The model integrates location and capacity choices for suppliers, plants and warehouses selection, product range assignment and production flows. The open-or-close decisions for the facilities are binary decision variables and the production and transportation flow decisions are continuous decision variables. Consequently, this problem is a binary mixed integer linear programming problem. In this paper, a modified version of Benders’ decomposition is proposed to solve the model. The most difficulty associated with the Benders’ decomposition is the solution of master problem, as in many real-life problems the model will be NP-hard and very time consuming. In the proposed procedure, the master problem will be developed using the surrogate constraints. We show that the main constraints of the master problem can be replaced by the strongest surrogate constraint. The generated problem with the strongest surrogate constraint is a valid relaxation of the main problem. Furthermore, a near-optimal initial solution is generated for a reduction in the number of iterations.     

Stochastic programming problems with generalized integrated chance constraints

If the constraints in an optimization problem are dependent on a random parameter, we would like to ensure that they are fulfilled with a high level of reliability. The most natural way is to employ chance constraints. However, the resulting problem is very hard to solve. We propose an alternative formulation of stochastic programs using penalty functions. The expectations of penalties can be left as constraints leading to generalized integrated chance constraints, or incorporated into the objective as a penalty term. We show that the penalty problems are asymptotically equivalent under quite mild conditions. We discuss applications of sample-approximation techniques to the problems with generalized integrated chance constraints and propose rates of convergence for the set of feasible solutions. We will direct our attention to the case when the set of feasible solutions is finite, which can appear in integer programming. The results are then extended to the bounded sets with continuous variables. Additional binary variables are necessary to solve sample-approximated chance-constrained problems, leading to a large mixed-integer non-linear program. On the other hand, the problems with penalties can be solved without adding binary variables; just continuous variables are necessary to model the penalties. The introduced approaches are applied to the blending problem leading to comparably reliable solutions.     

Allocation of dependent divisional resources

A resource allocation problem is considered with resources that are dependent in the sense that an allocation to an activity requires the application of several resources, except for certain activities which are divisional in the sense that an allocation to such an activity requires the use of only a single resource. Return and cost functions are assumed to be continuous and increasing, and the allocation variables are continuous. Conditions are given for the replacement of the continuous problem by an associated problem with discrete variables and a single constraint, and to a given degree of accuracy. The associated problem can be efficiently solved by dynamic programming. Certain divisional resource allocation problems with discrete variables and several linear constraints are shown to be equivalent to a discrete problem with a single constraint. A numerical example is given.     

Discrete and geometric Branch and Bound algorithms for?medical image registration

Aiming at the development of an exact solution method for registration problems, we present two different Branch & Bound algorithms for a mixed integer programming formulation of the problem. The first B&B algorithm branches on binary assignment variables and makes use of an optimality condition that is derived from a graph matching formulation. The second, geometric B&B algorithm applies a geometric branching strategy on continuous transformation variables. The two approaches are compared for synthetic test examples as well as for 2-dimensional medical data. The results show that medium sized problem instances can be solved to global optimality in a reasonable amount of time.     

Parking Capacity and Pricing in Park'n Ride Trips: A Continuous Equilibrium Network Design Problem

In this paper we consider the problem of designing parking facilities for park'n ride trips. We present a new continuous equilibrium network design problem to decide the capacity and fare of these parking lots at a tactical level. We assume that the parking facilities have already been located and other topological decisions have already been taken.The modeling approach proposed is mathematical programming with equilibrium constraints. In the outer optimization problem, a central Authority evaluates the performance of the transport network for each network design decision. In the inner problem a multimodal traffic assignment with combined modes, formulated as a variational inequality problem, generates the share demand for modes of transportation, and for parking facilities as a function of the design variables of the parking lots. The objective is to make optimal parking investment and pricing decisions in order to minimize the total travel cost in a subnetwork of the multimodal transportation system.We present a new development in model formulation based on the use of generalized parking link cost as a design variable.The bilevel model is solved by a simulated annealing algorithm applied to the continuous and non-negative design decision variables. Numerical tests are reported in order to illustrate the use of the model, and the ability of the approach to solve applications of moderate size.     

An exact penalty function method for nonlinear mixed discrete programming problems

In this paper, we consider a general class of nonlinear mixed discrete programming problems. By introducing continuous variables to replace the discrete variables, the problem is first transformed into an equivalent nonlinear continuous optimization problem subject to original constraints and additional linear and quadratic constraints. Then, an exact penalty function is employed to construct a sequence of unconstrained optimization problems, each of which can be solved effectively by unconstrained optimization techniques, such as conjugate gradient or quasi-Newton methods. It is shown that any local optimal solution of the unconstrained optimization problem is a local optimal solution of the transformed nonlinear constrained continuous optimization problem when the penalty parameter is sufficiently large. Numerical experiments are carried out to test the efficiency of the proposed method.     

Solving continuous min max problem for single period portfolio selection with discrete constraints by DCA

In this article, we consider the application of a continuous min–max model with cardinality constraints to worst-case portfolio selection with multiple scenarios of risk, where the return forecast of each asset belongs to an interval. The problem can be formulated as minimizing a convex function under mixed integer variables with additional complementarity constraints. We first prove that the complementarity constraints can be eliminated and then use Difference of Convex functions (DC) programming and DC Algorithm (DCA), an innovative approach in non-convex programming frameworks, to solve the resulting problem. We reformulate it as a DC program and then show how to apply DCA to solve it. Numerical experiments on several test problems are reported that demonstrate the accuracy of the proposed algorithm.     

Linear programming with variable matrix entries

We consider linear programming (continuous or integer) where some matrix entries are decision parameters. If the variables are nonnegative the problem can be easily solved in two phases. It is shown that direct costs on the matrix entries make the problem NP-hard. Finally, a strong duality result is provided.     

Global optimization of mixed-integer bilevel programming problems

Two approaches that solve the mixed-integer nonlinear bilevel programming problem to global optimality are introduced. The first addresses problems mixed-integer nonlinear in outer variables and C²-nonlinear in inner variables. The second adresses problems with general mixed-integer nonlinear functions in outer level. Inner level functions may be mixed-integer nonlinear in outer variables, linear, polynomial, or multilinear in inner integer variables, and linear in inner continuous variables. This second approach is based on reformulating the mixed-integer inner problem as continuous via its vertex polyheral convex hull representation and solving the resulting nonlinear bilevel optimization problem by a novel deterministic global optimization framework. Computational studies illustrate proposed approaches.     

Global optimization of signomial mixed-integer nonlinear programming problems with free variables

Mixed-integer nonlinear programming (MINLP) problems involving general constraints and objective functions with continuous and integer variables occur frequently in engineering design, chemical process industry and management. Although many optimization approaches have been developed for MINLP problems, these methods can only handle signomial terms with positive variables or find a local solution. Therefore, this study proposes a novel method for solving a signomial MINLP problem with free variables to obtain a global optimal solution. The signomial MINLP problem is first transformed into another one containing only positive variables. Then the transformed problem is reformulated as a convex mixed-integer program by the convexification strategies and piecewise linearization techniques. A global optimum of the signomial MINLP problem can finally be found within the tolerable error. Numerical examples are also presented to demonstrate the effectiveness of the proposed method.     

Two-Stage Stochastic Variational Inequality Arising from Stochastic Programming

We consider a two-stage stochastic variational inequality arising from a general convex two-stage stochastic programming problem, where the random variables have continuous distributions. The equivalence between the two problems is shown under some moderate conditions, and the monotonicity of the two-stage stochastic variational inequality is discussed under additional conditions. We provide a discretization scheme with convergence results and employ the progressive hedging method with double parameterization to solve the discretized stochastic variational inequality. As an application, we show how the water resources management problem under uncertainty can be transformed from a two-stage stochastic programming problem to a two-stage stochastic variational inequality, and how to solve it, using the discretization scheme and the progressive hedging method with double parameterization.

     

First-order inertial algorithms involving dry friction damping

We present a Branch-and-Cut algorithm for a class of nonlinear chance-constrained mathematical optimization problems with a finite number of scenarios. Unsatisfied scenarios can enter a recovery mode. This class corresponds to problems that can be reformulated as deterministic convex mixed-integer nonlinear programming problems with indicator variables and continuous scenario variables, but the size of the reformulation is large and quickly becomes impractical as the number of scenarios grows. The Branch-and-Cut algorithm is based on an implicit Benders decomposition scheme, where we generate cutting planes as outer approximation cuts from the projection of the feasible region on suitable subspaces. The size of the master problem in our scheme is much smaller than the deterministic reformulation of the chance-constrained problem. We apply the Branch-and-Cut algorithm to the mid-term hydro scheduling problem, for which we propose a chance-constrained formulation. A computational study using data from ten hydroplants in Greece shows that the proposed methodology solves instances faster than applying a general-purpose solver for convex mixed-integer nonlinear programming problems to the deterministic reformulation, and scales much better with the number of scenarios.

     

Global optimization for generalized geometric programming problems with discrete variables

Generalized geometric programming (GGP) problems occur frequently in engineering design and management, but most existing methods for solving GGP actually only consider continuous variables. This article presents a new branch-and-bound algorithm for globally solving GGP problems with discrete variables. For minimizing the problem, an equivalent monotonic optimization problem (P) with discrete variables is presented by exploiting the special structure of GGP. In the algorithm, the lower bounds are computed by solving ordinary linear programming problems that are derived via a linearization technique. In contrast to pure branch-and-bound methods, the algorithm can perform a domain reduction cut per iteration by using the monotonicity of problem (P), which can suppress the rapid growth of branching tree in the branch-and-bound search so that the performance of the algorithm is further improved. Computational results for several sample examples and small randomly generated problems are reported to vindicate our conclusions.