Piecewise Linear Programming Research Articles

Iterative schemes for the least 2-norm solution of piecewise linear programs

Let f : R n → (−∞, ∞] be a convex polyhedral function. We show how to find the normal minimizer of f and the associated Lagrange multipliers by computing x(ϵ) = arg min x f(x) + ϵ x 2 2 approximately for a sequence of ϵ ↓ 0 via any relaxation method applied to the corresponding dual problems. Our schemes generalize those of Managasarian and De Leone for solving very large sparse linear programs.

A Cholesky dual method for proximal piecewise linear programming

A quadratic programming method is given for minimizing a sum of piecewise linear functions and a proximal quadratic term, subject to simple bounds on variables. It may be used for search direction finding in nondifferentiable optimization algorithms. An efficient implementation is described that updates a Cholesky factorization of active constraints and provides good accuracy via iterative refinement. Numerical experience is reported for some large problems.

A multicriterion approach to groundwater management

A new approach to groundwater quantity (hydraulic) management, based on a multicriterion decision aid methodology, is presented. The method couples a hydrodynamic groundwater flow simulation model with the decision aid one. It takes into account multiple criteria used by the decision maker (DM) in evaluating a management scenario as well as the hydrodynamic behavior of the groundwater system. The finite element and embedding methods are used to integrate the groundwater flow model into a multiobjective linear programming (LP) problem. Constraints on head, pumping rates, hydraulic gradient and velocity vector may be included in the management model. The piecewise linear utility function is assessed for modeling the DM's preferences. The best compromise solution is determined from the continuous Pareto set by solving a piecewise LP problem. User friendly software was developed to realize this methodology which is able to treat real scale problems. An illustrative example of an unconfined aquifer management is presented. Nonlinearities resulting from functional dependence of aquifer parameters on hydraulic head are handled iteratively.

Alternative optimization strategies for large-scale production-allocation problems

Many optimization problems arising in practice are so complicated that the current tools for optimization do not supply a single, clean answer. We are forced to apply a variety of optimization methods on approximate models, so that we also obtain a variety of approximate solutions. Using a production-allocation problem in Philips Electronics Industries as a vehicle for communication, we generate several near-optimal solutions via decomposition, piece-wise linear programming, and mixed-integer programming. Next, we show that multi-criteria analysis could be used to select one of the resulting allocation patterns for implementation in practice. Thus, instead of a problem with a continuum of alternatives, we eventually consider a finite number of alternatives under several conflicting performance criteria. The study particularly illustrates the significance of mathematical modelling in the power game between globally operating product divisions and locally operating national organizations.

A generator and a simplex solver for network piecewise linear programs

This is a brief report on our recent work in network piecewise linear programming (NPLP), and it consists of two parts. In the first park, we describe a generator for NPLP problems which is derived from the classical network linear program generator NETGEN. The generator creates networks of the same topological structures as NETGEN, but each arc is associated with a convex piecewise linear cost. The purpose of this program is to provide a set of standard test problems which can be used to compare the performance of various algorithms for NPLP. In the second part, we introduce a network simplex method that directly solves a network piecewise linear program without reformulating it as a network linear program of higher dimension. Forty benchmark NPLP problems are solved by this method and a reformulation method. The computational results are in favor of the direct method and show that solving an NPLP problem is not much harder than solving a network linear program of the same dimension.

Optimal assignment of NOP due-dates and sequencing in a single machine shop

We consider the problem of optimal assignment of NOP due-dates ton jobs and sequencing them on a single machine to minimize a penalty function depending on the values of assigned constant waiting allowance and maximum job tardiness. It is shown that the earliest due date (EDD) order is an optimal sequence. For finding optimal constant waiting allowance, we reduce the problem to a multiple objective piecewise linear programming with single variable. An efficient algorithm for optimal solution of the problem is given.

Minimax refining of optical multilayer systems

A new minimax method for refining optical multilayer systems is presented. It minimizes the maximum deviation of the spectral transmittance from the desired specifications. These are assigned in such a general way that any shape can be approximated. The algorithm consists of iterating optimization steps that are obtained by developing the transmittance in the Taylor expansion versus the known parameters and solved by piecewise-linear programming. The investigation is limited to the design of losslessmultilayers with assigned refractive indices and undetermined thicknesses. Some minimax-refined designs are compared favorably with those reported in the literature.

Solving Piecewise-Linear Programs: Experiments with a Simplex Approach

We report tests of CPLP, a piecewise-linear simplex implementation based on the XMP subroutine library. Using test problems of various shapes and origins, we judge CPLP against a comparable linear bounded-variable simplex code that is applied to equivalent linear programs. Our evidence indicates that CPLP is typically 2 to 3 times faster in solving the test problems that are most significantly piecewise-linear; previously predicted advantages of the piecewise-linear simplex approach are observed to occur, though with different combinations of advantages being relevant in different cases. The savings achieved by CPLP appear likely to extend, moreover, to the costs involved in formulating, generating and analyzing piecewise-linear programs. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

A simplex algorithm for piecewise-linear programming III: Computational analysis and applications

The first two parts of this paper have developed a simplex algorithm for minimizing convex separable piecewise-linear functions subject to linear constraints. This concluding part argues that a direct piecewiselinear simplex implementation has inherent advantages over an indirect approach that relies on transformation to a linear program. The advantages are shown to be implicit in relationships between the linear and piecewise-linear algorithms, and to be independent of many details of implementation. Two sets of computational results serve to illustarate these arguments; the piecewise-linear simplex algorithm is observed to run 2–6 times faster than a comparable linear algorithm, not including any additional expense that might be incurred in setting up the equivalent linear program. Further support for the practical value of a good piecewise-linear programming algorithm is provided by a survey of many varied applications.

Implementing the modified LH algorithm

The modified Lemke-Howson algorithm is a constructive procedure which enables us to compute equilibrium points of a bimatrix game. The algorithm as described by one of the authors is based on the original version invented by Lemke and Howson. However, it differs from this version with respect to several features. It works directly with the matrices defining the bimatrix game A and B. It has an easy and very direct geometrical interpretation; hence for small games we can follow the development of the algorithm geometrically. Finally, instead of being bilinear, the algorithm behaves rather like a piecewise linear program. This presentation closes a gap: although the algorithm has been described geometrically (and with a flow diagram), there has been no constructive procedure that can be implemented on a computer. This is provided by the present paper. We give all necessary proofs and computations in order to establish the following facts: There are two tableaus accompanying the proceeding of the algorithm. As the algorithm changes, moving alternatingly in the simplices of mixed strategies, so does the computational procedure alternatingly dealing with the two different tableaus. Each tableau contains six regions, depending on the various sequences of transitions the procedure has to perform. While this all is in marked difference to linear programming, there is also consolation: The well-known rectangle rule of linear programming can be modified easily (that is, there is a family of rectangle rules) so that changing the tableau alternatingly amounts to applying the appropriate rectangle rule. Thus, there is also close similarity to the familiar LP procedure. Thus, a complete description of the modified LH algorithm is provided that can immediately be implemented on any computer. In particular, we supply an APL program that, e.g., can be run on an IBM ® PC.

SOA: Large Strain Consolidation Predictions

Large deformation consolidation computer programs are being used extensively by Florida phosphate mining industries for compliance with regulatory activities. Accordingly, a prediction symposium was held to provide a forum of idealized boundary conditions that would demonstrate areas of agreement and the limitations of various computer programs. Nine different program users provided predictions of pond elevation histories and of pore pressure and void ratio profiles after one year for four different waste clay disposal scenarios. Programs to estimate quiescent consolidation of waste ponds having uniform initial void ratios, with or without surcharge loading, and continuous and/or stage filling are available, but less available are programs that handle nonuniform void ratios and/or multilayer conditions with different nonlinear compressibility properties. Presently, piecewise linear programs appear best suited to accommodate the latter. In general, final height predictions for all scenarios were in agreeme...

A simplex algorithm for piecewise-linear programming II: Finiteness, feasibility and degeneracy

The simplex method for linear programming can be extended to permit the minimization of any convex separable piecewise-linear objective, subject to linear constraints. Part I of this paper has developed a general and direct simplex algorithm for piecewise-linear programming, under convenient assumptions that guarantee a finite number of basic solutions, existence of basic feasible solutions, and nondegeneracy of all such solutions. Part II now shows how these assumptions can be weakened so that they pose no obstacle to effective use of the piecewise-linear simplex algorithm. The theory of piecewise-linear programming is thereby extended, and numerous features of linear programming are generalized or are seen in a new light. An analysis of the algorithm's computational requirements and a survey of applications will be presented in Part III.

Application program for the Sulaibiya solar thermal power plant: Energy flow and water allocation model

The Sulaibiya solar thermal power system is designed for the decentralized supply of energy for small isolated communities. The variability in solar insolations makes it necessary to study operating options available and the corresponding allocation of energy and water to users. This article presents a mathematical programming model for the operation of the plant. A piecewise linear programming approach is adopted to reflect the variable efficiency of the turbines based on input levels. The optimization criterion is a linear combination of two objectives: maximizing solar fraction and satisfying users' demands. Relative weights are attached to convey the importance of each criterion. This model aims at integrating the energy and water components of the system, and providing daily operating options for various insolation levels and demand profiles. The model was validated at a design point assuming 100% solar insolation for summer demand profiles. Based on this validation, the model proved to be successful in representing the real situation, which encourages its use to provide different operating options under various conditions.

Piecewise-linear programming: The compact (CPLP) algorithm

Piecewise-linear programming: The compact (CPLP) algorithm

Piecewise-linear programming: The compact (CPLP) algorithm

A compact algorithm is presented for solving the convex piecewise-linear-programming problem, formulated by means of a separable convex piecewise-linear objective function (to be minimized) and a set of linear constraints. This algorithm consists of a finite sequence of cycles, derived from the simplex method, characteritic of linear programming, and the line search, characteristic of nonlinear programming. Both the required storage and amount of calculation are reduced with respect to the usual approach, based on a linear-programming formulation with an expanded tableau. The tableau dimensions arem×(n+1), wherem is the number of constraints andn the number of the (original) structural variables, and they do not increase with the number of breakpoints of the piecewise-linear terms constituting the objective function.

Successive approximation in separable programming: An improved procedure for convex separable programs

AbstractWe implement a solution procedure for general convex separable programs where a series of relatively small piecewise linear programs are solved as opposed to a single large one, and where, based on bound calculations developed in [13] and [14], the ranges of linearization are systematically reduced for successive programs. The procedure inherits ε‐convergence to the global optimum in a finite number of steps, but perhaps its most distinct feature is the rigorous way in which ranges containing an optimal solution are reduced from iteration to iteration. This paper describes the procedure, called successive approximation, discusses its convergence, tightness of the bounds, bound‐calculation overhead, and its robustness. It presents a computer implementation to demonstrate its effectiveness for general problems and compares it (1) with the more standard separable programming approach and (2) with one of the recent augmented Lagrangian methods [10] included in a comprehensive study of nonlinear programming codes [12]. It seems clear from over 130 cases resulting from 80 distinct problems studied here that significant savings in terms of computational effort can be realized by a judicious use of the procedure, and the ease with which it can be used is appreciably increased by the robustness it shows. Moreover, for most of these problems, the advantage increases as the size, nonlinearity, and the degree of desired accuracy increase. Other important benefits include significantly smaller storage requirements, the ability to estimate the error in the current solution, and to terminate the algorithm as soon as the acceptable level of accuracy has been achieved. Problems requiring up to about 10,000 nonzero elements in their specification and about 45,000 nonzero elements in the generated separable programs resulting from up to 70 original nonlinear variables and 70 nonlinear constraints are included in the computations.

DC solution speed in piecewise linear network analysis programs

The letter discusses the computer execution time for the DC solution of nonlinear networks using a particular analysis program, based on piecewise linear methods. A simple but effective technique is shown to considerably reduce the execution time; it is expected that the method could be applied to other programs based on PWL methods.

A simplex algorithm for piecewise-linear programming I: Derivation and proof

The simplex method for linear programming can be extended to permit the minimization of any convex separable piecewise-linear objective, subject to linear constraints. This three-part paper develops and analyzes a general, computationally practical simplex algorithm for piecewiselinear programming.

Applications of methods for solving of special nonlinear programming problems to optimization of control systems

The algorithms of optimization of dynamic control systems in the class of impulse and piece-wise constant functions are considered in the report. They are based on the methods for solving of linear, piece-wise linear and quadratic programming problems worked out by the authors. I. Consider the time optimal control problem(1)x˙=Ax+Bu,xO=xo,d*≤ut≤d*,t∈T-O,ti,(2)ti→min,xti=xi,where x=xt=xit,…xnt is the vector of state of the system at the moment t;A∈Rn×n,B,xo,xi∈Rn;rankB,AB,…,An−1b=n;u=ut,t∈T, is the one-dimensional admissible contro from the class of piece-wise constant functions that are continuous to the right at the points of discontinuity. The algorithm for solving of problem(l-2)is based on the primal support method for solving of linear problem of terminal control with the unbound termination of the process and on the use of differencial equations for optimal support and optimal value of cost functional in the problem being considered. A numerical experiment was carried out to test the efficiency of the suggested method. The problems of the typeti→min,x˙i=x2,x˙2=x3,…,x˙n=u,xO=xO,xti=xi,ut≤i,t∈T=O,ti,were chosen as the test ones where the order of the system n =3,5, the initial approximation to the time optimal moment, the initial and terminal states varied by special rules. The accuracy of 10-15-10-15 was reached. 2. Consider the terminal control problemc′xto→max,x˙t=Atxt+Btut+ft,t∈T=tN,tK;B*i≤h′ixti≤Bi*,i=l,m¯;d*t≤ut≤d*t,in the class of the impulse controls u(t) , where c,xt,ft∈Rn,ut∈R2,t∈TUnder the numerical solution of the given problem the trajectory xt,t∈T of the system is approximately computed and the accuracy varies from iteration to iteration. The algorithm is finite without taking into consideration that the accuracy of the calculations approaches zero. At the first stage the “rough” calculations are conducted. As approaching to the optimum according to the definite rules the accuracy is raising. This allows to economize GPU time. The numerical experiment showed that the rate of algorithm increases approximately two times in comparison with the variant where the maximal accuracy is constantly used at iterations. 3. Three problems are considered for the system (I):I) Ju=minc′ixti+αi→max, 2)Ju=∑i∈Kc′ixti+αi→min, 3)Ju=maxi∈Kc′ixti+αi→min,with the constraintsHxti=g,f*t≤ut≤f*t,t∈T,where K is the finite set, ci are the vectors, i∈K;αi are scalars i∈K. The exact finite algorithms are worked out. They are programmed in FORTRAN-17. 4. The problem of quadratic functional on the trajectories of the system (2) is investigated.

PLMAP — A piecewise linear MOS circuit analysis program

In this paper we describe a MOS circuit simulation program PLMAP ( Piecewise Linear MOS Circuit Analysis Program) which performs efficient time domain simulation of LSI MOS circuits. PLMAP uses a piecewise linear model for MOS transistors. It uses a relaxation method to analyze the piecewise linear circuit at one time point after another. The relaxation method consists of a 2-level decomposition which partitions the whole circuit into 2-level subcircuits and a scheduling method which determines the sequence of analysis of subcircuits. At each time point, the subcircuits are analyzed sequentially and iteratively by using Gauss-Seidel iteration together with a prediction formula to determine the initial guess of solution. Experiences have shown that PLMAP is more efficient in MOS circuit analysis than spice.