APROS: Algorithmic Development Methodology for Discrete-Continuous Optimization Problems

Abstract

APROS represents an algorithmic development procedure for classes of mathematical programming problems that involve some form of decomposition technique and require extensive communication of data between a set of subproblems whose sizes and structures may vary during the solution procedure. Examples include most classes of mixed-integer nonlinear programming problems and large-scale mixed-integer linear programming problems as well as a wide variety of algorithms for large-scale nonlinear and linear programming problems exhibiting special structure. APROS works through the General Algebraic Modeling System (GAMS) to provide exact syntactic statement of algorithmic solution procedures. APROS procedures are implemented through GAMS which is interfaced with nonlinear, linear and mixed-integer linear programming solvers to provide completely general automated implementations of many well known algorithms including the Generalized Benders Decomposition, the Outer Approximation/Equality Relaxation and Dantzig-Wolfe Decomposition. The flexibility of APROS and the highly procedural modeling language GAMS provide a means of obtaining quick solutions to difficult classes of problems by implementing a selected algorithm, and also provide a unique tool for developing, prototyping and experimenting with new algorithms. A description of the procedural components, features and setups is presented with emphasis on the generality of the techniques.

A two‐step method for mixed‐integer nonlinear programming problems

“…The focus of this paper is mixed-integer nonliner programming problems, which are illustrated in Paules and Floudas ( [1], pp. 911-912).…”

“…229-232) and Wong ([6], [7], and [8]), a computer program for carrying out this step one of selecting an acceptable penalty coefficient and for solving the first example problem of Paules and Floudas [1] was run. (Like the other computer programs of this paper, this program can be obtained from the author.)…”

“…This second step was carried out for the first example problem of Paules and Floudas [1] by changing the program mentioned above into a new program that only uses 50 as the penalty coefficient. (All computer programs of this paper are readily available by writing to the author.)…”

“…Modelled after the two programs mentioned above, two computer programs for solving the second example problem in Paules and Floudas [1] were run. The first program's purpose was to find an acceptable penalty coefficient; the second program's purpose was to find a solution vector for the example problem.…”

Integrating network optimization capabilities into a high-level modeling language

“…This is especially true in the context of large-scale models, where we wish to investigate several algorithmic alternatives quickly before investing in the development of software for large-scale computations. This arrangement has been shown, Paules and Floudas [18], to be a suitable vehicle for the automated solution of mixed integer and nonlinear programs.…”