# András PrékopaRutgers, The State University of New Jersey | Rutgers · Statistics

András Prékopa

PhD

## About

138

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3,726

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Citations since 2016

## Publications

Publications (138)

Given a stochastic process with known finite dimensional distributions, we construct lower and upper bounds within which future values of the stochastic process run, at a fixed probability level. For a financial trading business, such set of bounds are called “price-bands” or “trading-bands” that can be used as an indicator for successfully buying...

Clustering interval data has been studied for decades. High-dimensional interval data can be expressed in terms of hyperrectangles in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \b...

Lower and upper bounds are derived on single-period European options under moment information, without assuming that the asset prices follow geometric Brownian motion, which is frequently untrue in practice. Sometimes the entire asset distribution is not completely known, sometimes it is known but the numerical calculation is easier by the use of t...

Probability that exactly k-out-of-n events occur, where the underlying probability distribution is known to be unimodal, is formulated as binomial moment problems. Dual feasible basis structures of the corresponding linear programs are fully described. Closed form bounds based on the first two binomial moments are presented. A dual simplex method b...

We consider probabilistically constrained stochastic programming problems, in which the random variables are in the right-hand sides of the stochastic inequalities defining the joint chance constraints. Problems of that kind arise in a variety of contexts, and are particularly difficult to solve for random variables with continuous joint distributi...

In the first two sections of the paper, stream flow is investigated on a probability theoretical basis. We will show that under some realistic conditions its probability distribution is of gamma type. In the model of the third section the optimal capacity of a storage reservoir is determined. In the model of the fourth section optimal water release...

New multivariate risk measures are introduced, suitable for optimal management of multidimensional assets. Risk is measured along lines through a given reference point in a multidimensional Euclidean space, and then maximum (minimum in financial planning) or mixture is taken with respect to lines lying in cones. We use VaR and CVaR as univariate ri...

The contribution of the shape information of the underlying distribution in probability bounding problem is investigated and a linear programming based bounding methodology to obtain robust and efficiently computable bounds for the probability that at least -out-of- events occur is developed. The dual feasible basis structures of the relaxed versio...

We propose a novel optimization model to find reliable bounds of a real valued function of a simple random sample from a population. If the sample size is , then the function would be a function of i.i.d. random variables (by simple random sampling). Comprehensive evaluation by simulation is challenging when the function is asymmetric, requiring a...

In this paper, our sets are orthants in and , the number of them, is large ( ). We introduce the modified inclusion-exclusion formula in order to efficiently calculate the probability of a union of such events. The new formula works in the bivariate case, and can also be used in with a condition on the projected sets onto lower dimensional spaces....

We present a brief survey of some of the basic results related to the classical continuous moment problems (CMP) and the recently developed discrete moment problems (DMP), clarifying their relationship and propose new methods for the solution of univariate continuous and discrete power moment problems. In the classical as well as in the recently de...

Single commodity networks are considered, where demands at the nodes are random. The problem is to find minimum cost optimal built in capacities at the nodes and arcs subject to the constraint that all demands should be met on a prescribed probability level (reliability constraint) and some deterministic constraints should be satisfied. The reliabi...

A new numerical integration method, termed Discrete Moment Method, is proposed for univariate functions that are piecewise higher order convex. This means that the interval where the function is defined can be subdivided into non-overlapping subintervals such that in each interval all divided differences of given orders, do not change the sign. The...

Let A(1),..., A(N) and B-1,..., B-M be two sequences of events and let v(N) (A) and v(M) (B) be the number of those A(i) and B-j, respectively, that occur. Based on multivariate Lagrange interpolation, we give a method that yields linear bounds in terms of S-k,S-t, k + t <= m on the distribution of the vector (v(N) (A), v(M) (B)). For the same valu...

We formulate and solve probabilistic constrained stochastic programming problems, where we prescribe lower and upper bounds for k-out-of- n and consecutive- k-out-of- n reliabilities in the form of probabilistic constraints. Problems of this kind arise in many practical situations where stockout in a number of consecutive or a certain fraction of a...

In this chapter a general mathematical programming model of the scheduling problem as formulated in Sect. 1.2 is presented for the case where the electric power generation system includes thermal plants only. The model is called general because no simplifying assumptions are applied for the sake of obtaining a mathematical programming model tractab...

Both the generation system and the network transmitting electric power vary on a day-to-day basis; therefore, the daily optimization problems to be solved and their sizes also vary, although their structure remains essentially the same. This chapter shows how the daily scheduling problem, corresponding to the simplified model of the scheduling prob...

On the basis of the discussion in Chap. 3, the general model of the scheduling problem can be simplified.
The simplified model is a large-scale mixed-variable mathematical programming problem with a linear objective function and linear constraints, with the coefficient matrix having a special structure. It is suitable for numerical solutions.

An electric power system is a combination of power-producing units, transmission lines, international cooperation, transformers, and a distribution network supplying customers with power under joint supervision and control.

In Chap. 2, our aim in formulating the general model of the scheduling problem was to construct a model that would best describe the problem, regardless of whether we had any chance of solving the corresponding mathematical programming problem. According to the overview of the model in Sect. 2.4, the corresponding problem is a large-scale, mixed-va...

Corporate Mergers and Acquisitions (M \( { \& }\) As) are notoriously complex, and risk management is one of the essential aspects of the analysis process for decision-making on M \( { \& }\) A deals. Empirically, we see that some M \( { \& }\) A transactions are not successful in part because of the increased exposure to correlated sectors, sugges...

A recent paper by Prékopa (Ann. Oper. Res. 193(1):49–69, 2012) presented results in connection with Multivariate Value-at-Risk (MVaR) that has been known for some time under the name of p-quantile or p-Level Efficient Point (pLEP) and introduced a new multivariate risk measure, called Multivariate Conditional Value-at-Risk (MCVaR). The purpose of t...

Compound distributions come up in many applications (telecommunication, hydrology, insurance, etc.), where some of the typical problems are of optimization type. The log-concavity property is paramount in these respects to ensure convexity. In this paper, we prove the log-concavity of some compound Poisson and other compound distributions.

Discrete moment problems (DMP) with integer moments were first introduced by Prékopa to provide sharp lower and upper bounds for functions of discrete random variables. Prékopa also developed fast and stable dual type linear programming methods for the numerical solutions of the problem. In this paper, we assume that some fractional moments are als...

The following question arises in stochastic programming: how can one approximate a noisy convex function with a convex quadratic
function that is optimal in some sense. Using several approaches for constructing convex approximations we present some optimization
models yielding convex quadratic regressions that are optimal approximations in L
1, L
∞...

A. Pr�ekopa and E. Boros published a paper \On the Existence of a Feasible Flow
in a Stochastic Transportation Network" that appeared in Operations Research
39(1991), 119-129. In that paper the authors gave a powerful technique to evalu-
ate transportation system reliability. At the nodes production takes place and the
products are shipped on the a...

Multivariate Value at Risk, or MVaR, is defined as the quantile set of a multivariate probability distribution. It has already
been introduced and used in the literature under the name of p-Level Efficient Points, or pLEP’s, or briefly p-efficient points. Some of the topics connected with it are surveyed: discrete convexity, algorithmic generation,...

Joshi and Staunton [Quantit. Finance, 2012, 12, 17–20] have commented on the paper by Prékopa and Szántai [Quantit. Finance, 2010, 10, 59-74] and criticized the statement that the binomial tree method overestimates the option price, under some condition. In this paper we present our more detailed reasoning that clarifies some property of the mechan...

Different kinds of stochastic programming models are formulated in the present mathematical programming literature. Their solutions lead to linear or non-linear deterministic programming problems. There are, however, a number of problems, mainly probability distribution problems, which remained unsolved which are nevertheless important and necessar...

A probabilistic constrained stochastic linear programming problem is considered, where the rows of the random technology matrix are independent and normally distributed. The quasi-concavity of the constraining function needed for the convexity of the problem is ensured if the factors of the function are uniformly quasi-concave. A necessary and suff...

The paper further develops, both from the theoretical and numerical points of view the analytical valuation of the American options, initiated by Geske and Johnson (198420.
Geske , R and
Johnson , HE . 1984. The American put option valued analytically. J. Finan., 39: 1511–1524. [CrossRef], [Web of Science ®]View all references) for the American p...

Discrete moment problems with given finite supports and unimodal distributions with known mode, are formulated and used to obtain sharp lower and upper bounds for expectations of higher order convex functions of discrete random variables as well as probabilities of the union of events. The bounds are based on the knowledge of some of the power mome...

A dual type linear programming algorithm is presented for locating the maximum of a strongly unimodal multivariate discrete distribution.

The models discussed in the present paper are generalizations of the models introduced previously by A. Prékopa [6] and M.
Ziermann [13]. In the mentioned papers the initial stock level of one basic commodity is determined provided that the delivery
and demand process allow certain homogeneity (in time) assumptions if they are random. Here we are d...

Mathematically a natural river system is a rooted directed tree where the orientations of the edges coincide with the directions
of the streamflows. Assume that in some of the river valleys it is possibie to build reservoirs the purpose of which will
be to retain the flood, once a year, say. The problem is to find optimal reservoir capacities by mi...

A discrete function f defined on Zn is said to be logconcave if f(λx+(1−λ)y)≥[f(x)]λ[f(y)]1−λ for x, y, λx+(1−λ)y∈Zn. A more restrictive notion is strong unimodality. Following Barndorff-Nielsen [O. Barndorff-Nielsen, Unimodality and exponential families, Commun. Statist. 1 (1973) 189–216] a discrete function p(z),z∈Zn is called strongly unimodal i...

Prékopa (1973, 1995) has proved that if T is an r ×n random matrix with independent, normally distributed rows such that their covariance matrices are constant multiples of each other, then the function h(x) = P (T x ≥ b) is quasi-concave in R n , where b is a constant vector. We prove that, under same condition, the converse is also true, a specia...

A function u(z) is a utility function if u′(z) > 0. It is called risk averse if we also have u′′(z) < 0. Some authors, however, require that u
(i)(z) > 0 if i is odd and u
(i)(z) < 0 if i is even. The notion of a multiattribute utility function can be defined by requiring that it is increasing in each variable and concave as an s-variate function....

Linear programming problem is formulated for bounding the probability of the union of events, where the probability distribution of the occurrences is supposed to be unimodal with known mode and some of the binomial moments of the events are also known. Using a theorem on combinatorial determinants the dual feasible bases of a relaxed problem are f...

Keywords
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References

When formulating a stochastic programming problem, we usually start from a deterministic problem that we call underlying deterministic problem. Then, observing that some of the parameters are random, we formulate another problem, the stochastic programming problem, by taking into account the probability distribution of the random elements in the un...

In this chapter we look at the following problems $$
Max\quad G(x) = P({g_1}(x,\xi ) \geqslant 0,{g_2}(x,\xi ) \geqslant 0,...,{g_r}(x,\xi ) \geqslant 0)\quad {\text{subject to }}{h_1}(x) \geqslant {p_{1,}}{h_2}(x) \geqslant {p_2},...,{h_m}(x) \geqslant {p_m}{\text{ }}
$$ (10.1.1)
and Min h(x) Subject to
$$
G(x) = P({g_1}(x,\xi ) \geqslant 0,{g_2}(...

Multivariate probability distributions with given marginals are considered, along with linear functionals, to be minimized or maximized, acting on them. The functionals are supposed to satisfy the Monge or inverse Monge or some higher order convexity property, and they may be only partially known. Existing results in connection with Monge arrays ar...

A probabilistic constrained stochastic programming model is formulated, where one term in the objective function, to be minimized, is the maximum of a finite or infinite number of linear functions. The model is reformulated as a finite or semi-infinite disjunctive programming problem. Duality relationships are established for both the original and...

The paper shows how the bounding technique provided by the multivariate discrete moment problem can be used for bounding expectations of functions of random variables with known univariate marginals and some of the mixed moments. Four examples are presented. In the first one the function is a Monge or related type array, in the second one it is a p...

In this paper we recall and further develop an inventory model formulated by the author [Prékopa, A., 1965. Reliability equation for an inventory problem and its asymptotic solutions. In: Prékopa, A. (Ed.), Colloquia Applied Mathematics in Economics. Publ. House of the Hung. Acad. Sci., Budapest, pp. 317–327; Prékopa, A., 1973. Generalizations of t...

We consider stochastic programming problems with probabilistic constraints involving integer-valued random variables. The concept of p-efficient points of a probability distribution is used to derive various equivalent problem formulations. Next we modify the concept of r-concave discrete probability distributions and analyse its relevance for prob...

This paper gives a rigorous definition of a stage, usable for dynamic stochastic programs with both recourse and probabilistic constraints. Algebraic modelling languages can make use of this definition for automatic consistency checks.

Given a sequence of n arbitrary events in a probability space, we assume that the individual probabilities as well as some or all joint probabilities of up to m events are known, 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...

The discrete moment problem (DMP) has been formulated as a methodology to find the minimum and/or maximum of a linear functional acting on an unknown probability distribution, the support of which is a known discrete (usually finite) set, where some of the moments are known. The moments may be binomial, power, or of more general type. The multivari...

An LP is considered where the technology coefficients are unknown and random samples are taken to estimate them. A stochastic programming problem is formulated to find the optimal sample sizes where it is required that a confidence interval should cover the unknown deterministic optimum value by a given probability and the cost of sampling be minim...

A project is defined as the collection of activities (or events) {a, 6,…} among which a precedence relation a ≺ b is defined. It is supposed to be transitive, i.e., if a ≺ b and b ≺c then a ≺c. Any project can be depicted as a directed network, where the directed arcs represent the activities. Without restricting
generality, we may assume that ther...

The purpose of this paper is twofold. First to present a brief survey of some of the basic results related to the univariate moment problem, including Prékopa's dual approach for solving the discrete moment problem. Second we propose a new method for solving the continuous power moment problem when some higher order divided differences of the objec...

We consider stochastic integer programming problems with probabilistic constraints. The concept of p-efficient points of a probability distribution is used to derive various equivalent problem formulations. Next we introduce new methods for constructing lower and upper bounds for probabilistically constrained integer programs. We also show how limi...

This article revises and improves on a Dual Type Method (DTM), developed by Prékopa. (Prékopa, A. (1990). Dual method for the solution of a one-stage stochastic programming problem with random RHS obeying a discrete probability distribution, ZOR-Methods and Models of Operations Research , 34 , 441-461), in two ways. The first one allows us, in each...

We prove that the multivariate standard normal probability distribution function is concave for large argument values. The method of proof allows for the derivation of similar statements for other types of multivariate probability distribution functions too. The result has important application, e.g., in probabilistic constrained stochastic program...

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 at-...

In this paper we present an overview about the recently developed theory of discrete moment problems, i.e., moment problems
where the supports of the random variables involved are discrete. We look for the minimum or maximum of a linear functional
acting on an unknown probability distribution subject to a finite number of moment constraints. Using...

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...

Stochastic programming models of optical fiber production planning are presented. The purpose is to set the optimal fiber manufacturing goals while accounting for the uncertainty primarily in the yield and secondly in the demand. The model is solved for the case when the data follows a multivariate discrete distribution, and also for the case of a...

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...

In the past few years, efficient methods have been developed for bounding probabilitiesand expectations concerning univariate and multivariate random variables based on theknowledge of some of their moments. Closed form as well as algorithmic lower and upperbounds of this type are now available. The lower and upper bounds are frequently closeenough...

The most important static stochastic programming models, that can be formulated in connection with a linear programming problem, where some of the right-hand side values are random variables, are: the simple recourse model, the probabilistic constrained model and the combination of the two. In this paper, we present algorithmic solution to the seco...

This didactic paper presents a brief, algorithm-oriented mathematical theory of linear programming including the simplex method, the dual method, their lexicographic variants, finiteness, duality theory, Farkas’ theorem, von Neumann’s theorem on games and Gomory’s all integer algorithm. Results from Dantzig, Lemke, Charnes, Gale, Kuhn and Tucker an...

A dual method is presented to solve a linearly constrained optimization problem with convex, polyhedral objective function, along with a fast bounding technique, for the optimum value. The method can be used to solve problems, obtained from LPs, where some of the constraints are not required to be exactly satisfied but are penalized by piecewise li...

Events that occur consecutively or simultaneously cause some other event as effect. The latter can be observed with noise, and the problem is to estimate the weights of the causes in the realization of the effect.

Many transportation networks, e.g., networks of cooperating power systems, and hydrological networks involve a real-valued demand function, defined on the set of nodes, and it is said to be feasible if there exists a flow such that at each node the sum of the incoming flow values is greater than or equal to the demand assigned to this node. By the...

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 pen...

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 Lagr...

In a previous paper (1988), the author proposed methods to obtain sharp lower and upper bounds for probabilities that at least one out of n events occurs, based on the knowledge of some of the binomial moments of the number of events which occur and linear programming formulations. This paper presents further results concerning other and more gener...

In this paper we present a new method to analyze and solve the maximum satisfiability problem. We randomize the Boolean variables, assign probabilities to their possible values and, by using recently developed probabilistic bounds of the authors, present a deterministic procedure to obtain solution to the maximum satisfiability problem. Our algorit...