András Prékopa

Concavity and Efficient Points of Discrete Distributions in Probabilistic Programming

Abstract.We consider stochastic programming problems with probabilistic constraints involving integer-valued random variables. The concept of a p-efficient point of a probability distribution is used to derive various equivalent problem formulations. Next we introduce the concept of r-concave discrete probability distributions and analyse its relevance for problems under consideration. These notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables. The results are illustrated with numerical examples.

Contributions to the theory of stochastic programming

Two stochastic programming decision models are presented. In the first one, we use probabilistic constraints, and constraints involving conditional expectations further incorporate penalties into the objective. The probabilistic constraint prescribes a lower bound for the probability of simultaneous occurrence of events, the number of which can be infinite in which case stochastic processes are involved. The second one is a variant of the model: two-stage programming under uncertainty, where we require the solvability of the second stage problem only with a prescribed (high) probability. The theory presented in this paper is based to a large extent on recent results of the author concerning logarithmic concave measures.

Closed form two-sided bounds for probabilities that at least r and exactly r out of n events occur

In two previous papers Prekopa (Prekopa, A. 1986a. Boole-Bonferroni inequalities and linear programming. Oper. Res. 36 145–162; Prekopa, A. 1986b. Sharp bounds on probabilities using linear programming. To appear in Oper. Res.) gave algorithms to approximate probabilities that at least r and exactly r out of n events occur (1 ≤ r ≤ n). Primal and dual linear programming problems were formulated and solved by dual type algorithms. The purpose of the present paper is to give closed forms for the basis inverse and the corresponding dual vector in case of an arbitrary basis, furthermore to give closed forms for the lower and upper bounds, approximating the probability in question, in case of a dual feasible basis. In the case when the probability that at least one out of n events occurs is approximated, it is shown that the absolute values of the components of any dual vector form a monotonically decreasing sequence. The paper improves the method of inclusion-exclusion, proves new probability inequalities and...

Dual method for the solution of a one-stage stochastic programming problem with random RHS obeying a discrete probability distribution

In this paper we present a method for the solution of a one stage stochastic programming problem, where the underlying problem is an LP and some of the right hand side values are random variables. The stochastic programming problem that we formulate contains probabilistic constraint and penalty, incorporated into the objective function, used to penalize violation of the stochastic constraints. We solve this problem by a dual type algorithm. The special case where only penalty is used while the probabilistic constraint is disregarded, the simple recourse problem, was solved earlier by Wets, using a primal simplex algorithm with individual upper bounds. Our method appears to be simpler. The method has applications to nonstochastic programming problems too, e.g., it solves the constrained minimum absolute deviation problem.ZusammenfassungIn dieser Arbeit wird eine Methode vorgestellt zur Lösung einstufiger stochastischer Programme, wobei das zugrundeliegende Problem ein LP mit zufälligen rechten Seiten darstellt. Das resultierende stochastische Programm enthält Wahrscheinlichkeitsrestriktionen und Strafterme, letztere innerhalb der Zielfunktion zur Bestrafung von Abweichungen in den stochastischen Restriktionen. Wir lösen dieses Problem mit einem dualen Algorithmus. Der Spezialfall, in dem ausschließlich Strafterme benutzt werden und Wahrscheinlichkeitsrestriktionen unberücksichtigt bleiben, d.h. das einfache Kompensationsmodell, wurde bereits früher von Wets mittels eines primalen Simplex-Algorithmus mit einzelnen oberen Schranken gelöst. Unsere Methode scheint einfacher zu sein. Die Methode ist auch auf nicht-stochastische Programme anwendbar, z.B. auf das Problem minimaler absoluter Abweichungen von Nebenbedingungen.

Programming Under Probabilistic Constraint with Discrete Random Variable

The most important static stochastic programming models, that can b€ formulated in connection with a linear programming problem, where some of th€ right-hand side values are random variables, are: the simple recourse model, th€ probabilistic constrained model and the combination of the two. In this paper w€ present algorithmic solution to the second and third models under the assumption that the random variables have a discrete joint distribution. The solution method uses the concept of a p-Ievel efficient point (pLEP) intoduced by the first author (1990) and works in such a way that first all pLEPs are enumerated, then a cutting plane method does the rest of the job.

The discrete moment problem and linear programming

Abstract Moment problems, with finite, preassigned support, regarding the probability distribution, are formulated and used to obtain sharp lower and upper bounds for unknown probabilities and expectations of convex functions of discrete random variables. The bounds are optimum values of special linear programming problems. Simple derivations, based on Lagrange polynomials, are presented for the dual feasible basis structure theorems in case of the power and binomial moment problems. The sharp bounds are obtained by dual type algorithms and formulas. They are analoguous to the Chebyshev-Markov inequalities.

Probability Bounds with Cherry Trees

A third order upper bound is presented on the probability of the union of a finite number of events, by means of graphs called cherry trees. These are graphs that we construct recursively in such a way that every time we pick a new vertex, connect it with two already existing vertices. If the latters are always adjacent, we call the cherry tree a t-cherry tree. A cherry tree has a weight that provides us with the upper bound on the union. Any Hunter-Worsley bound can be majorized by a t-cherry bound constructed by the use of the Hunter-Worsley tree. A cherry tree bound can be identified as a feasible solution to the dual of the Boolean probability bounding problem. A t-cherry tree bound can be identified as the objective function value of the dual vector corresponding to a dual feasible basis in the Boolean problem. This enables us to improve on the bound algorithmically, if we use the dual method of linear programming.

PROGRAMMING UNDER PROBABILISTIC CONSTRAINTS WITH A RANDOM TECHNOLOGY MATRIX

Probabilistic constraint of the type P (Ax ≦ β) ≧ p is proved that under some conditions the constraining function is quast-conceave. The probabilistic constraint is embedded into a mathematical programming problem of which the algorithmie solution is also discussed.

ON THE PROBABILITY DISTRIBUTION OF THE OPTIMUM OF A RANDOM LINEAR PROGRAM

it is also a random variable and its probability distribution is what we are interested in. This problem is of basic importance and is conceivable as a stochastic sensitivity analysis of a linear programming model. The question how the transformation A, b, c → μ operates under the presence of random influences in A, b, and c does not play just the role of a sensitivity analysis, however. In fact, in A, b, c we may have not just small random disturbances but random variables of significant variation.

Bounding the probability of the union of events by aggregation and disaggregation in linear programs

Given a sequence of n arbitrary events in a probability space, we assume that the individual probabilities as well some or all joint probabilities of up to m events are know, where m<n. Using this information we give lower and upper bounds for the probability of the union. The bounds are obtained as optimum values of linear programming problems or objective function values corresponding to feasible solutions of the dual problems. If all joint probabilities of the k-tuples of events are known, for k not exceeding m, then the LP is the large-scale Boolean probability bounding problem. Another type of LP is the binomial moment problem, where we assume the knowledge of some of the binomial moments of the number of events which occur. The two LPs can be obtained from each other by aggregation/disaggregation procedure In this paper, we define LPs which are obtained by partial aggregation/disaggregation from these two LPs. This way we can keep the size of the problem low but can produce very good bounds in many cases. The obtained lower bounds generalize the bounds of de Caen (Discrete Math. 169 (1997) 217) and Kuai, Alajaji and Takahara (Discrete Appl. Math. 215 (2000) 147). Practical applications are mentioned and numerical examples are presented.