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To solve an integer linear program, we identify particular values that the objective function can assume on feasible lattice points. Thus, we reduce the problem of finding an optimal integer solution of n dimensions to that of finding a feasible integer solution of n-1 dimensions. A Branch and Bound Method is presented to solve the latter problem for the 0-1 case.

A parametric integer linear program (PILP) may be defined as a family of closely related integer linear programs (ILP). Within this definition the author incorporates not only continuous scalar parameterizations but also finite parameterizations. These may include an ILP with a finite number of objective functions or right hand sides or constraint matrices or any combination of these. A general framework for PILP is presented. It begins by outlining the need for PILP algorithms. Basic solution methodologies are explained and two rudimentary approaches for the PILP are stated. Theoretical properties for special parameterizations are proved, and techniques for improving algorithmic efficiency are discussed. The framework concludes with an examination of underlying factors which intimately relate to the scheduling of solution priorities in a PILP algorithm.

We consider the following problem: Given a rational matrix $A \in \setQ^{m \times n}$ and a rational polyhedron $Q \subseteq\setR^{m+p}$, decide if for all vectors $b \in \setR^m$, for which there exists an integral $z \in \setZ^p$ such that $(b, z) \in Q$, the system of linear inequalities $A x \leq b$ has an integral solution. We show that there exists an algorithm that solves this problem in polynomial time if $p$ and $n$ are fixed. This extends a result of Kannan (1990) who established such an algorithm for the case when, in addition to $p$ and $n$, the affine dimension of $Q$ is fixed. As an application of this result, we describe an algorithm to find the maximum difference between the optimum values of an integer program $\max \{c x : A x \leq b, x \in \setZ^n \}$ and its linear programming relaxation over all right-hand sides $b$, for which the integer program is feasible. The algorithm is polynomial if $n$ is fixed. This is an extension of a recent result of Ho\c{s}ten and Sturmfels (2003) who presented such an algorithm for integer programs in standard form.

Bibliography: leaf 26

Bibliography: leaf 26

Bibliography: leaf 26

In recent years, the embedding approach for switched optimal control problems has been developed in a series of papers. However, the embedding approach, which advantageously converts the hybrid optimal control problem to a classical nonlinear optimization, has not been extensively compared to alternative approaches. The goal of this paper is thus to compare the embedding approach to multi-parametric programming, mixed-integer programming, gradient-descent based methods, and CPLEX in the context of five recently published examples. A sixth example, an autonomous switched 11-region linear system, is used to compare a hybrid minimum principle method and traditional numerical programming. For a given performance index for each case, cost and solution times are presented. It is shown that there are numerical advantages of the embedding approach: lower performance index cost (except in some instances when autonomous switches are present), generally faster solution time, and convergence to a solution when other methods may fail. In addition, the embedding method requires no ad hoc assumptions (e.g., predetermined mode sequences) or specialized control models. Theoretical advantages of the embedding approach over the other methods are also described: guaranteed existence of a solution under mild conditions, convexity of the embedded hybrid optimization problem (under the customary conditions on the performance index), solvability with traditional techniques (e.g., sequential quadratic programming) avoiding the combinatorial complexity in the number of modes/discrete variables of mixed-integer programming, applicability to affine nonlinear systems, and no need to explicitly assign discrete/mode variables to autonomous switches.

Postoptimality analysis and parametric optimization techniques are fully developed aspects of linear programming. In the context of integer linear programming, however, these aspects have barely begun to be developed. The purpose of this paper is to take stock of what is known about this topic and to lay the foundation for future progress. Our conceptual starting point is the notion that, in practical applications, typically one is faced not with a single numerical integer linear program to solve but rather with an entire family of numerical problems of interest. The members of the family may all have the same structure but differ as to the values of one or more coefficients, or they may even have different (but related) structures. The scope of this paper is limited to be first mentioned case.