Random linear programs with many variables and few constraints

We extend and simplify Smale's work on the expected number of pivots for a linear program with many variables and few constraints. Our analysis applies to new versions of the simplex algorithm and to new random distributions.     

Zero-one integer programs with few constraints —Lower bounding theory

The zero-one integer programming problem and its special case, the multiconstraint knapsack problem frequently appear as subproblems in many combinatorial optimization problems. We present several methods for computing lower bounds on the optimal solution of the zero-one integer programming problem. They include Lagrangean, surrogate and composite relaxations. New heuristic procedures are suggested for determining good surrogate multipliers. Based on theoretical results and extensive computational testing, it is shown that for zero-one integer problems with few constraints surrogate relaxation is a viable alternative to the commonly used Lagrangean and linear programming relaxations. These results are used in a follow up paper to develop an efficient branch and bound algorithm for solving zero-one integer programming problems.     

Zero-one integer programs with few constraints —Efficient branch and bound algorithms

The 0–1 integer programming problem and its special case, the 0–1 knapsack problem are frequently encountered in modeling various design and decision making processes. This paper is a follow-up paper to [4] and deals with the development of an effective solution procedure for 0–1 integer programs with few constraints. Detailed computational experiments are carried out and different separation, branching and bounding rules are compared using an experimental branch and bound code. An efficient branch and bound procedure is developed, tested and compared with previously developed optimal algorithms. It is suggested that this procedure may also be used as a heuristic method for large problems by early termination of the tree search. This scheme is tested and found to be very effective.     

Solving nonlinear programming problems with very many constraints

For solving the smooth constrained nonlinear programming problem, sequential quadratic programming (SQP) methods are considered to be the standard tool, as long as they are applicable. However one possible situation preventing the successful solution by a standard SQP-technique, arises if problems with a very large number of constraints are to be solved. Typical applications are semi-infinite or min-max optimization, optimal control or mechanical structural optimization. The proposed technique proceeds from a user defined number of linearized constraints, that is to be used internally to determine the size of the quadratic programming subproblem. Significant constraints are then selected automatically by the algorithm. Details of the numerical implementation and some experimental results are presented     

Flattened aggregate function method for nonlinear programming with many complicated constraints

Numerical Algorithms - In this paper, efforts are made to solve nonlinear programming with many complicated constraints more efficiently. The constrained optimization problem is firstly converted...     

Group automorphisms with few and with many periodic points

For each a compact group automorphism is constructed with the property that

This may be interpreted as a combinatorial analogue of the (still open) problem of whether compact group automorphisms with any given topological entropy exist.

     

Indestructibility and measurable cardinals with few and many measures

If κ < λ are such that κ is indestructibly supercompact and λ is measurable, then we show that both A = {δ < κ | δ is a measurable cardinal which is not a limit of measurable cardinals and δ carries the maximal number of normal measures} and B = {δ < κ | δ is a measurable cardinal which is not a limit of measurable cardinals and δ carries fewer than the maximal number of normal measures} are unbounded in κ. The two aforementioned phenomena, however, need not occur in a universe with an indestructibly supercompact cardinal and sufficiently few large cardinals. In particular, we show how to construct a model with an indestructibly supercompact cardinal κ in which if δ < κ is a measurable cardinal which is not a limit of measurable cardinals, then δ must carry fewer than the maximal number of normal measures. We also, however, show how to construct a model with an indestructibly supercompact cardinal κ in which if δ < κ is a measurable cardinal which is not a limit of measurable cardinals, then δ must carry the maximal number of normal measures. If we weaken the requirements on indestructibility, then this last result can be improved to obtain a model with an indestructibly supercompact cardinal κ in which every measurable cardinal δ < κ carries the maximal number of normal measures. A. W. Apter’s research was partially supported by PSC-CUNY grants and CUNY Collaborative Incentive grants. In addition, the author wishes to thank the referee, for helpful comments, corrections, and suggestions which have been incorporated into the current version of the paper.     

An economical method of calculating Lagrange multiplier estimates for nonlinear programming problems when many of the constraints are bounds on the variables

Some recent methods for solving nonlinear programming problems make use of estimates of the Lagrange multipliers. These estimates are usually calculated by solving a system oft linear equations, wheret is the number of active constraints. It is shown that, when a large proportion of the active constraints consists of simple upper or lower bounds on the variables, then computational effort can be saved by means of a reorganization of this linear system.     

Toda flows with infinitely many variables

The Toda flow and related flows extend naturally to operators in Hilbert space and the purpose of this paper is to describe these flows and to analyse some of their special properties.     

The splitting of variables and constraints in the formulation of integer programming models

The splitting of variables in an integer programming model into the sum of other variables can allow the constraints to be disaggregated, leading to a more constrained (tighter) linear programming relaxation. Well known examples of such reformulations are quoted from the literature. They can be viewed as instances of some general methods of performing such reformulations, namely disjunctive formulations, partial network reformulations and a method based on the introduction of auxiliary variables.     

Optimization with additional variables and constraints

Norton, Plotkin and Tardos proved that—loosely spoken, an LP problem is solvable in time O(Tq^k+1) if deleting k fixed columns or rows, we obtain a problem which can be solved by an algorithm that makes at most T steps and q comparisons. This paper improves this running time to O(Tq^k).     

MARKOV DECISION PROGRAMMING WITH CONSTRAINTS

ＭＡＲＫＯＶＤＥＣＩＳＩＯＮＰＲＯＧＲＡＭＭＩＮＧＷＩＴＨＣＯＮＳＴＲＡＩＮＴＳＬＩＵＪＩＡＮＹＯＮＧ（刘建庸）；ＬＩＵＫＥ（刘克）（ＩｎｓｔｉｔｕｔｅｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ，ｔｈｅＣｈｉｎｅｓｅＡｃａｄｅｍｙｏｆＳｃｉｅｎｃｅｓ，...     

Markov programming with policy constraints

In the theory and applications of Markov decision processes introduced by Howard and subsequently developed by many authors, it is assumed that actions can be chosen independently at each state. A policy constrained Markov decision process is one where selecting a given action in one state restricts the choice of actions in another. This note describes a method for determining a maximal gain policy in the policy constrained case. The method involves the use of bounds on the gain of the feasible policies to produce a policy ranking list. This list then forms a basis for a bounded enumeration procedure which yields the optimal policy.     

On geometric optimization with few violated constraints

We investigate the problem of finding the best solution satisfying all butk of the given constraints, for an abstract class of optimization problems introduced by Sharir and Welzl—the so-calledLP-type problems. We give a general algorithm and discuss its efficient implementations for specific geometric problems. For instance for the problem of computing the smallest circle enclosing all butk of the givenn points in the plane, we obtain anO(n logn+k ³ n ^ε) algorithm; this improves previous results fork small compared withn but moderately growing. We also establish some results concerning general properties ofLP-type problems. This research was supported in part by Charles University Grant No. 351 and Czech Republic Grant GAČR 201/93/2167. Part of this research was performed while the author was visting the Computer Science Institute, Free University Berlin, and it was supported by the German-Israeli Foundation of Scientific Research and Development (G.I.F.), and part while visiting the Max-Planck Institute for Computer Science in Saarbrücken.     

Stochastic programming with integer variables

 Including integer variables into traditional stochastic linear programs has considerable implications for structural analysis and algorithm design. Starting from mean-risk approaches with different risk measures we identify corresponding two- and multi-stage stochastic integer programs that are large-scale block-structured mixed-integer linear programs if the underlying probability distributions are discrete. We highlight the role of mixed-integer value functions for structure and stability of stochastic integer programs. When applied to the block structures in stochastic integer programming, well known algorithmic principles such as branch-and-bound, Lagrangian relaxation, or cutting plane methods open up new directions of research. We review existing results in the field and indicate departure points for their extension. Received: December 2, 2002 / Accepted: April 23, 2003 Published online: May 28, 2003 Mathematics Subject Classification (2000): 90C15, 90C11, 90C06, 90C57     

Random inequality constraint systems with few variables

We show that a system of many linear inequality constraints will have a high proportion of redundant constraints with high probability. Implications for expected time of algorithms are indicated.     

Linear programming with matrix variables

Linear programming is formulated with the vector variable replaced by a matrix variable, with the inner product defined using trace of a matrix. The theorems of Motzkin, Farkas (both homogeneous and inhomogeneous forms), and linear programming duality thus extend to matrix variables. Duality theorems for linear programming over complex spaces, and over quaternion spaces, follow as special cases.