I We define the complete problem of a two-stage linear programming under uncertainty v to bet Minimize zlx) =E-{cx+qy + q"y" } subject to Ax = b Tx + Iy + + ly" ■ K I x^Oy +^o y"^0 where x is the first-stege decision varisble, the peir (y iy") represents the second-stage decision variables* In order to solve this class of problem, we derive a convex programming problem, whose set of optimal solutions is identical to the set of optimal solutions of our original problen-. This problem is called

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... e equiyalent convex programming« If the random variable I has a continuous distribution, we give an algorithm to solve the equivalent convex program. Moreover, we derive explicitly the equivalent convex program for a few common distributions* ■■.-•■ i .M' ... k- (2) z( x) = ex + E^ { min qy I x } . All quantities considered here belong to the reals, denoted R . Vectors will belong to finite-dimensional real vector spaces R and whether they are to be regarded as row vectors or column vectors will always be clear from , the context in which they appear. Thus, for example» the expressions X s \Xi$• • • tX^f• • • $X£) Tx= x 4. -ffi i=i are easily understood. No special provisions will be made for transposing vectors. The random vector C = (5, •••• Cj 5-) is a "numerical" 1 1 m random vector, i.e. Z c R m » 0 is an algebra or a o -algebra and F is a probability distribution function from which could be obtained a probability measure. & » B*» F.) is the probability space of the random variable C. • We only need independence of C and x t our first-stage decision has then no effect on (J,8,F) . If for every finite interval, '«(C*) has a finite number of discontinuity points, then we can always integrate by parts | 8 i (^4) d F i (^i) » where gj(Cj) is a linear function of K. . If it exists, we denote the density function of K. by f^j) i = l,...,m and let a. and ß. be respectively the greatest lower bound and least upper bound, if they exist, of K, . We assume that £. { (. } exists for all 1=1, m. *i 1 We say that problem (1) is complete when the matrix M (after an appropriate rearrangement of rows and columns) can be partitioned in two parts, whose first part is an identity matrix and the second part is the negative of an identity itrix, M« (I, -I) • The standard form of th« problem to bt studied In thla article is then (3) Minimize a(x) » IL{cx + q + y + 4 q"y"] subject to Ax B b Tx + I7 + -ly" « C Ctd, 3, F) x 1 0,7 + ^ 0,y" ^ 0 where we partitioned the vectors q and 7 of the standard form (1) in (q ,q~) and (7 ,y") f respective^« The fact that m = 0 (I.e. there are no constraints of type Ax ^ b) does not alter the characteristics of our problem* Among all classes of special cases of the two-stage linear programs under uncertainty, the "complete 11 case seems to cover the largest class of possible applications« One can think of the vector x as representing the activity levels of a production plant, constrained by Ax = b , x ^ 0 . T is the "transfomation" of these activity levels into sellable goods» x c Tx, is then the amount of goods the producer decides to place on a market where the demand, 5 , is only known in probability« y and y" represent the "errors" the producer made in estimating the demand) q and q~ are penalty costs for making these "errors". For instance, an inventory type problem has T ■ I ,q represents the unit shortage cost, and q~ the unit holding oost, and Ax = b the capacity, budget, technology,••« constraints« It can be shown that the correlations between the %. do not enter the problem) we do not need thf independence of the C.» Ua denote the marginal distribution functions by F.(0 1 B l,»«... 9 m« • . -