We analyze the joint extremal behavior of n random products of the form φm j=1 X aij j , 1 ≤ i ≤ n, for non-negative, independent regularly varying random variables X1, . . . , Xm and general coefficients aij € R. Products of this form appear for example if one observes a linear time series with gamma type innovations at n points in time. We combine arguments of linear optimization and a generalized concept of regular variation on cones to show that the asymptotic behavior of joint exceedance probabilities of these products is determined by the solution of a linear program related to the matrix A = (aij ).