Linear, integer separable and fuzzy programming problems: a united approach towards automatic reformulation

For mathematical programming (MP) to have greater impact as a decision tool, MP software systems must offer suitable support in terms of model communication and modelling techniques. In this paper modelling techniques that allow logical restrictions to be modelled in integer programming terms are described and their implications discussed. In addition it is demonstrated that many classes of non-linearities which are not variable separable may be after suitable algebraic manipulation put in a variable separable form. The methods of reformulating the fuzzy linear programming problem as a Max-Min problem is also introduced. It is shown that analysis of bounds plays a key role in the following four important contexts: model reduction, reformulation of logical restrictions as 0-1 mixed integer programs, reformulation of nonlinear programs as variable separable programs and reformulation of fuzzy linear programs. It is observed that as well as incorporating an interface between the modeller and the optimiser there is a need to make available to the modeller software facilities which support the model reformulation techniques described here

Computer assisted modelling of linear, integer and separable programming problems

For mathematical programming (MP) to have greater impact upon the decision making process, MP software systems must offer suitable support in terms of model communication and modelling techniques . In this paper modelling techniques that allow logical restrictions to be modelled in integer programming terms are described and their implications discussed. In addition it is demonstrated that many classes of non-linearities which are not variable separable may be reformulated in piecewise linear form. It is shown that analysis of bounds is necessary in the following three important contexts: model reduction, formulation of logical restrictions as 0-1 mixed integer programs and reformulation of nonlinear programs as variable separable programs, It is observed that as well as incorporating an interface between the modeller and the optimiser there is a need to make available to the modeller software facilities which support the modelling techniques described here

Transformation of propositional calculus statements into integer and mixed integer programs: An approach towards automatic reformulation

A systematic procedure for transforming a set of logical statements or logical conditions imposed on a model into an Integer Linear Progamming (ILP) formulation Mixed Integer Programming (MIP) formulation is presented. An ILP stated as a system of linear constraints involving integer variables and an objective function, provides a powerful representation of decision problems through a tightly interrelated closed system of choices. It supports direct representation of logical (Boolean or prepositional calculus) expressions. Binary variables (hereafter called logical variables) are first introduced and methods of logically connecting these to other variables are then presented. Simple constraints can be combined to construct logical relationships and the methods of formulating these are discussed. A reformulation procedure which uses the extended reverse polish representation of a compound logical form is then described. These reformulation procedures are illustrated by two examples. A scheme of implementation.ithin an LP modelling system is outlined

Tools for reformulating logical forms into zero-one mixed integer programs (MIPS)

A systematic procedure for transforming a set of logical statements or logical conditions imposed on a model into an Integer Linear Programming (ILP) formulation or a Mixed Integer Programming (MIP) formulation is presented. A reformulation procedure which uses the extended reverse polish representation of a compound logical form is then described. A prototype user interface by which logical forms can be reformulated and the corresponding MIP constructed and analysed within an existing Mathematical Programming modelling system is illustrated. Finally, the steps to formulate a discrete optimisation model in this way are demonstrated by means of an example

A computational analysis of lower bounds for big bucket production planning problems

In this paper, we analyze a variety of approaches to obtain lower bounds for multi-level production planning problems with big bucket capacities, i.e., problems in which multiple items compete for the same resources. We give an extensive survey of both known and new methods, and also establish relationships between some of these methods that, to our knowledge, have not been presented before. As will be highlighted, understanding the substructures of difficult problems provide crucial insights on why these problems are hard to solve, and this is addressed by a thorough analysis in the paper. We conclude with computational results on a variety of widely used test sets, and a discussion of future research

A Practical Guide to Robust Optimization

Robust optimization is a young and active research field that has been mainly developed in the last 15 years. Robust optimization is very useful for practice, since it is tailored to the information at hand, and it leads to computationally tractable formulations. It is therefore remarkable that real-life applications of robust optimization are still lagging behind; there is much more potential for real-life applications than has been exploited hitherto. The aim of this paper is to help practitioners to understand robust optimization and to successfully apply it in practice. We provide a brief introduction to robust optimization, and also describe important do's and don'ts for using it in practice. We use many small examples to illustrate our discussions

Using a conic bundle method to accelerate both phases of a quadratic convex reformulation

We present algorithm MIQCR-CB that is an advancement of method MIQCR~(Billionnet, Elloumi and Lambert, 2012). MIQCR is a method for solving mixed-integer quadratic programs and works in two phases: the first phase determines an equivalent quadratic formulation with a convex objective function by solving a semidefinite problem $(SDP)$, and, in the second phase, the equivalent formulation is solved by a standard solver. As the reformulation relies on the solution of a large-scale semidefinite program, it is not tractable by existing semidefinite solvers, already for medium sized problems. To surmount this difficulty, we present in MIQCR-CB a subgradient algorithm within a Lagrangian duality framework for solving $(SDP)$ that substantially speeds up the first phase. Moreover, this algorithm leads to a reformulated problem of smaller size than the one obtained by the original MIQCR method which results in a shorter time for solving the second phase. We present extensive computational results to show the efficiency of our algorithm