Separable Nonlinear Programming Research Articles

On solving two-stage distributionally robust disjunctive programs with a general ambiguity set

We introduce two-stage distributionally robust disjunctive programs (TSDR-DPs) with disjunctive constraints in both stages and a general ambiguity set for the probability distributions. The TSDR-DPs subsume various classes of two-stage distributionally robust programs where the second stage problems are non-convex programs (such as mixed binary programs, semi-continuous program, nonconvex quadratic programs, separable non-linear programs, etc.). TSDR-DP is an optimization model in which the degree of risk aversion can be chosen by decision makers. It generalizes two-stage stochastic disjunctive program (risk-neutral) and two-stage robust disjunctive program (most-conservative). To our knowledge, the foregoing special cases of TSDR-DPs have not been studied until now. In this paper, we develop decomposition algorithms, which utilize Balas’ linear programming equivalent for deterministic disjunctive programs or his sequential convexification approach within L-shaped method, to solve TSDR-DPs. We present sufficient conditions under which our algorithms are finitely convergent. These algorithms generalize the distributionally robust integer L-shaped algorithm of Bansal et al. (SIAM J. on Optimization 28: 2360-2388, 2018) for TSDR mixed binary programs, a subclass of TSDR-DPs. Furthermore, we formulate a semi-continuous program (SCP) as a disjunctive program and use our results for TSDR-DPs to solve general two-stage distributionally robust SCPs (TSDR-SCPs) and TSDR-SCP having semi-continuous inflow set in the second stage.

A heuristical solution method to separable nonlinear programming problems

There are many methods for the solution of nonlinear programming (NLP) problems. This paper presents a new method, a heuristic, for the solution of large-scale separable NLP-problems. In this paper, separable NLP-problems are referred to a problem structure where each variable, in the problem, is only found in terms with a single variable. The method can tackle separable MINLP-problems as well. The proposed method is used to solve some smaller examples in order to show the usefulness of it on real problems.

A heuristical solution method to separable nonlinear programming problems

There are many methods for the solution of nonlinear programming (NLP) problems. This paper presents a new method, a heuristic, for the solution of large-scale separable NLP-problems. In this paper, separable NLP-problems are referred to a problem structure where each variable, in the problem, is only found in terms with a single variable. The method can tackle separable MINLP-problems as well. The proposed method is used to solve some smaller examples in order to show the usefulness of it on real problems.

WISCHE: A DSS for water irrigation scheduling

In this paper we present the models and the algorithms which are being used in a decision support system (DSS) to determine water irrigation scheduling. The DSS provides dynamic scheduling of the daily irrigation for a given land area by taking into account the irrigation network topology, the water volume technical conditions and the logistical operations. The system has been validated by the Agriculture Community of Elche (Spain) and annexed to their Supervisory Control and Data Acquisition system (SCADA). We present two heuristic approaches to solve the mixed 0–1 separable nonlinear program for irrigation scheduling implemented with free software.

On a mixed zero–one separable non-linear approach for water irrigation scheduling

We present a mixed zero–one separable non-linear approach to the optimization of the management of water resources used for agricultural irrigation purposes. It provides dynamic planning of the daily irrigation scheduling for a given land area by considering the irrigation network topography, water flow conditions and logistical operation constraints in order to optimize the use of water stored in a reservoir. We present a mixed zero–one separable non-linear program and a solution procedure that iteratively solves a mixed zero–one linear approximation of the model. We are not aware of any previous attempt to solve large-scale mixed zero–one separable non-linear programs of this kind. Some computational experiences on a large-scale real-life problem are reported.

On Decomposition Methods for a Class of Partially Separable Nonlinear Programs

We study two different decomposition algorithms for the general (nonconvex) partially separable nonlinear program (PSP): bilevel decomposition algorithms (BDAs) and Schur interior-point methods (SIPMs). BDAs solve the problem by breaking it into a master problem and a set of independent subproblems, forming a type of bilevel program. SIPMs, on the other hand, apply an interior-point technique to solve the problem in its original (integrated) form, but then use a Schur complement approach to solve the Newton system in a decentralized manner. Our first contribution is to establish a theoretical relationship between these two types of decomposition algorithms. This is a first step toward closing the gap between the incipient local convergence theory of BDAs and the mature local convergence theory of interior-point methods. Our second contribution is to show how SIPMs can be modified to solve problems for which the Schur complement matrix is not invertible in general. The importance of this contribution is that it substantially enlarges the class of problems that can be addressed with SIPMs.

On mixed-integer zero-one representations for separable lower-semicontinuous piecewise-linear functions

In a recent paper, Padberg (Oper. Res. Lett. 27(1) (2000) 1) has provided some insights into constructing locally ideal formulations for continuous piecewise-linear approximations to separable nonlinear programs. He shows that in contrast with such representations, the standard text-book modeling strategy is weak with respect to its linear programming relaxation. We propose a new pedagogically simpler modification of the latter formulation that constructs its convex hull representation, thereby rendering it locally ideal. Moreover, this modeling strategy imparts a totally unimodular structure to the formulation, it readily extends to representing separable lower-semicontinuous piecewise-linear functions, and it also facilitates a reduced locally ideal representation based on a piecewise-linear convex decomposition of the function. In the special case of continuous piecewise-linear functions, we exhibit a nonsingular linear transformation that equivalently converts the proposed model into Padberg's formulation.

Separable Quadratic Programming via a Primal-Dual Interior Point Method and its Use in a Sequential Procedure

This paper extends a primal-dual interior point procedure for linear programs to the case of convex separable quadratic objectives. Included are efficient procedures for: attaining primal and dual feasibility, variable upper bounding, and free variables. A sequential procedure that invokes the quadratic solver is proposed and implemented for solving linearly constrained convex separable nonlinear programs. Computational results are provided for several large test cases from stochastic programming. The proposed methods compare favorably with MINOS, especially for the larger examples. The nonlinear programs range in size up to 8,700 constraints and 22,000 variables. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

A Piecewise Method for Optimal Load-Flow Problem

Power utilities are increasingly being interconnected to improve the reliability of power supply and achieve economy of operation. Optimal load-flow analysis of such interconnected power systems, in general, results in complex large scale nonlinear programming problem, the direct method of solution of which requires huge amounts of core and computer time. Decomposition techniques have been extensively investigated in the power system literature to alleviate these difficulties. A piecewise method based on graph theoretic concept is proposed in this paper. The decomposition of the optimal load-flow into an additively separable nonlinear programming problems is achieved through the artifact of equivalent networks which were introduced in the piecewise load-flow problem. Lasdon’s decomposition technique is adapted for its solution. The optimization of subsystems is carried out using a shifted penalty function algorithm imbedded with a recent Fletcher’s unconstrained function minimization routine. Few examples are presented to demonstrate the effectiveness of this approach over the others.

A mathematical programming system for preference and compatibility maximized menu planning and scheduling

In this paper, the analytical representation of food preference is used in a separable non-linear program to yield the serving frequencies of menu items for a finite time horizon. The frequencies obtained in this way insure cost and nutritional control. Subsequently, the scheduling problem dealing with item assignments to meals and days is formulated as an integer program consisting of several transportation problems linked by weekly nutritional constraints. This problem is solved using a branch and bound algorithm which employs Lagrangian relaxation to obtain bounds and to decide on branching strategy.

A Linear Programming Analogue, a Duality Theorem, and a Dynamic Algorithm

This paper considers fluid analogues for the standard linear programming problem and for a separable nonlinear programming problem. In the former case the usual duality results are demonstrated using the principle of minimum potential energy. In addition by examining the dynamics of the system a new method, referred to as the R-method, is derived for solving linear programmes, although it is not demonstrated that this has any computational advantages over the standard simplex method, except that degeneracy causes no problems. In the nonlinear case the weak Lagrangian principle is derived. The purpose of the paper is to demonstrate that analogue methods, while being impracticable as a physical method of solving optimisation problems, may give some insight into computational algorithms via dual energy concepts and/or dynamic behaviour.

The Solution of Nonlinear Separable Programs

AbstractIt is shown that separable nonlinear programming problems may be solved approximately by using existing quadratic programming algorithms. The approach is to approximate the gradient functions by segmented linear functions which implies the approximation of the objective function by a smooth series of quadratic functions. A numerical example illustrates the technique.

On Programming under Uncertainty

The duality theory of programming under uncertainty is developed for the special case of proportional penalties and normal probability distributions, and from this an approximately optimal sampling plan is determined by the solution to a separable nonlinear programming problem with linear constraints. The duality theory is interesting per se; in particular, it evidences clearly the specific roles of the means and standard deviations in determining the solution. The duality theory of programming under uncertainty has been available for some time, at least in principle, by application of the general theory. It is worthwhile, however, to examine specially structured problems in finer detail, and that is the purpose of this paper.