Moment Problems and Semide Nite Optimization

Moment Problems and Semidenite Optimization y z Dimitris Bertsimas Ioana Pop escu Jay Sethuraman April Abstract Problems involving moments of random variables arise naturally in many areas of mathematics economics and op erations research Howdowe obtain optimal b ounds on the probability that a random variable b elongs in a set given some of its moments Howdowe price nancial derivatives without assuming any mo del for the underlying price dynamics given only moments of the price of the underlying asset Howdoweob tain stronger relaxations for sto chastic optimization problems exploiting the knowledge that the decision variables are moments of random variables Can we generate near optimal solutions for a discrete optimization problem from a semidenite relaxation by interpreting an optimal solution of the relaxation as a covariance matrix In this pap er we demonstrate that convex and in particular semidenite optimization metho ds lead to interesting and often unexp ected answers to these questions Dimitris Bertsimas Sloan Scho ol of Management and Op erations ResearchCenter Rm E MIT Cambridge MA db ertsimmitedu Research supp orted by NSF grant DMI and the Singap oreMIT alliance y Ioana Pop escu Decision Sciences INSEAD Fontainebleau Cedex FRANCE ioanap op escuin seadfr z Jay Sethuraman Columbia University DepartmentofIEORNewYork NY jayccolumbiaedu Intro duction Problems involving moments of random variables arise naturally in many areas of mathe matics economics and op erations research Let us give some examples that motivate the present pap er Moment problems in probability theory The problem of deriving b ounds on the probability that a certain random variable b elongs in a set given information on some of the moments of this random variable has a rich history whichisvery much connected with the development of probability theory in the twentieth century The inequalities due to Markov Chebyshev and Cherno are some of the classical and widely used results of mo dern probability theory Natural questions arise however a Are such bounds best possible ie do there exist distributions that match them A concrete and simple question in the univariate case Is the Chebyshev inequality best possible b Can such bounds be generalized in multivariate settings c Can we develop a general theory based on optimization methods to address moment problems in probability theory Moment problems in nance A central question in nancial economics is to nd the price of a derivative security given information on the underlying asset This is exactly the area of the Nob el prize in economics to Rob ert Merton and Myron Scholes Under the assumption that the price of the underlying asset follows a geometric Brownian motion and using the noarbitrage assumption the Blac kScholes formula provides an explicit and insightful answer to this question Natural questions arise however Making no assumptions on the underlying price dynamics but only using the noarbitrage assumption a What arethebest possible bounds for the price of a derivative security based on the rst and second moments of the price of the underlying asset b How can we derive optimal bounds on derivative securities that arebased on multiple underlying assets given the rst two moments of the asset prices and their correla tions c Conversely given observable option prices what arethebest bounds that we can derive on the moments of the underlying asset d Final ly given observable option prices what arethebest bounds that we can derive on prices of other derivatives on the same asset Moment problems in sto chastic optimization Scheduling a multiclass queueing network is a central problem in sto chastic optimization Queueing networks represent dynamic and sto chastic generalizations of job shops and have b een used in the last thirtyyears to mo del communication computer and manufacturing systems The central optimization problem is to nd a scheduling p olicy that optimizes a p erformance cost function c x d y where x x x x is the mean number of N j jobs of class j y y y and y is the second moment of the numb er of jobs of N j class j and c d are N v ectors of nonnegative constants The design of optimal p olicies is EX P T I M Ehard Papadimitiou and Tsitsiklis ie it provably requires exp onential time as P EX P T I M E A natural question that arises Can we nd strong lower bounds eciently exploiting the fact that the performancevectors represent moments of random variables Moment problems in discrete optimization The development of semidenite relaxations in recentyears represents an imp ortant ad vance in discrete optimization In several problems semidenite relaxations are provably closer to the discrete optimization solution value Go emans and Williamson for the maxcut problem for example than linear ones The pro of of closeness of the semidenite relaxation to the discrete optimization solution value involves a randomized argument that exploits the geometry of the semidenite relaxation A key question arises Is there a general methodofgenerating near optimal integer solutions starting from an op timal solution of the semidenite relaxation We will see that the interpretation of the solution of the semidenite relaxation as a co variance matrix for a collection of random variables leads to such a metho d and connects moment problems and discrete optimization The central message in this survey pap er is to demonstrate that convex and in partic ular semidenite optimization metho ds giveinteresting and often unexp ected answers to moment problems arising in probability economics and op erations research We also rep ort new computational results in the area of sto chastic optimization that show the eectiveness of semidenite relaxations The key connection The key connection b etween moment problems and semidenite optimization is centered in the notion of a feasible moment sequenceLetk k k bea vector of nonnegative n integers is a feasible n k moment vector or se Denition Asequence k k k k n n quence if there is a multivariate random variable X X X with domain R n k k n whose moments are given by k k k Wesay X thatis E X n k n that any such multivariate random variable X has a feasible distribution and denote this as X We denote by M Mn k the set of feasible n k momentvectors The univariate case For the univariate case n the problem of deciding if M M M isa k feasible kmomentvector is the classical moment problem which has b een completely characterized by necessary and sucient conditions see Karlin and Shapley Akhiezer Siu Sengupta and Lind and Kemp erman Theorem a Nonnegative random variables The vector M M is a n feasible n R moment sequence if and only if the fol lowing matrices are semide nite M M n C B C B C B M M M n C B R C B n C B C B A M M M n n n M M M n C B C B C B M M M n C B R C B n C B C B A M M M n n n b Arbitrary random variables The vector M M M is a feasible n R n moment sequence if and only if R n Pro of We will only show the necessity ofpartbIfM M M is a feasible n R n moment sequence then there exists a probability measure f xsuch that Z k x f xdx M k n k n where M Consider the vector x xx x and the semidenite matrix xx Since f x is nonnegative the matrix Z R xx f xdx n should b e semidenite The multivariate case n We consider the question of whether a sequence M is a feasible n R moment vector ie whether there exists a random vector X such that E X MEXX n Theorem Asequence M is a feasible n R moment vector if and only if the fol lowing matrix is semidenite M M Pro of n Supp ose M is a feasible n R momentvector Then there exists a random variable X such that E X MEXX The matrix X MX M is semidenite Taking exp ectations we obtain that E X MX M MM which expresses the fact that a covariance matrix needs to b e semidenite It is easy to see that MM if and only if Converselyif then MM LetX be a multivariate normal distribution ariance matrix MM This shows that the vector M is with mean M and cov n a feasible n R momentvector n There are known necessary conditions for a sequence to b e a feasible n k R moment vector for k that also involve the semideniteness of a matrix derived from the vector but these conditions are not known to b e sucient In general the complexity of deciding n whether a sequence is a feasible n k R momentvector has not b een resolved Structure of the pap er The structure of the pap er is as follows In Section we outline the application of semide nite programming to sto chastic optimization problems In Section we derive explicit and often surprising optimal b ounds in probability theory using convex and semidenite pro gramming metho ds In Section we apply convex and semidenite programming metho ds to problems in nance In Section we illustrate a connection b etween moment problems and semidenite relaxations in discrete optimization Section contains some concluding remarks Semidenite Relaxations for Sto chastic Optimization Prob lems The development of semidenite relaxations represents an imp ortant advance in discrete optimization In this section we review a theory for deriving semidenite relaxations for classical sto chastic optimization problems The idea of deriving semidenite relaxations for this class of problems is due to Bertsimas Our development in this pap er follows Bertsimas and NinoMora the interested reader is referred to these pap ers for fur ther

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