We consider the one-to-one Pickup and Delivery Problem (PDP) in Euclidean Space with arbitrary dimension @math where @math transportation requests are picked i.i.d. with a separate origin-destination pair for each object to be moved. First, we consider the problem from the customer perspective where the objective is to compute a plan for transporting the objects such that the Euclidean distance traveled by the vehicles when carrying objects is minimized. We develop a polynomial time asymptotically optimal algorithm for vehicles with capacity @math for this case. This result also holds imposing LIFO constraints for loading and unloading objects. Secondly, we extend our algorithm to the classical single-vehicle PDP where the objective is to minimize the total distance traveled by the vehicle and present results indicating that the extended algorithm is asymptotically optimal for a fixed vehicle capacity if the origins and destinations are picked i.i.d. using the same distribution.
Stein @cite_6 also conducts a probabilistic analysis but he looks at the variant where @math and @math and where @math origins and @math destinations are picked independently using a uniform distribution on some planar region. Stein shows that the value of the optimal solution divided by @math converges almost surely to a constant times the square root of the area of the region. Stein also shows how to solve the problem he considers by concatenating two TSP tours on the origins and destinations respectively. This way of solving the problem yields a solution which is roughly @math O(n^2) @math 4 @math d=2 @math c=+ @math d=2 @math c = O( n) @math d @math c$ @cite_13 .
@cite_13 @cite_6
2090424605 2083391552
The problem of finding large average submatrices of a real-valued matrix arises in the exploratory analysis of data from a variety of disciplines, ranging from genomics to social sciences. In this paper we provide a detailed asymptotic analysis of large average submatrices of an @math Gaussian random matrix. The first part of the paper addresses global maxima. For fixed @math we identify the average and the joint distribution of the @math submatrix having largest average value. As a dual result, we establish that the size of the largest square sub-matrix with average bigger than a fixed positive constant is, with high probability, equal to one of two consecutive integers that depend on the threshold and the matrix dimension @math . The second part of the paper addresses local maxima. Specifically we consider submatrices with dominant row and column sums that arise as the local optima of iterative search procedures for large average submatrices. For fixed @math , we identify the limiting average value and joint distribution of a @math submatrix conditioned to be a local maxima. In order to understand the density of such local optima and explain the quick convergence of such iterative procedures, we analyze the number @math of local maxima, beginning with exact asymptotic expressions for the mean and fluctuation behavior of @math . For fixed @math , the mean of @math is @math while the standard deviation is @math . Our principal result is a Gaussian central limit theorem for @math that is based on a new variant of Stein's method. Classical stochastic gradient methods are well suited for minimizing expected-value objective functions. However, they do not apply to the minimization of a nonlinear function involving expected values or a composition of two expected-value functions, i.e., the problem @math minxEvfv(Ew[gw(x)]). In order to solve this stochastic composition problem, we propose a class of stochastic compositional gradient descent (SCGD) algorithms that can be viewed as stochastic versions of quasi-gradient method. SCGD update the solutions based on noisy sample gradients of @math fv,gw and use an auxiliary variable to track the unknown quantity @math Ewgw(x). We prove that the SCGD converge almost surely to an optimal solution for convex optimization problems, as long as such a solution exists. The convergence involves the interplay of two iterations with different time scales. For nonsmooth convex problems, the SCGD achieves a convergence rate of @math O(k-1 4) in the general case and @math O(k-2 3) in the strongly convex case, after taking k samples. For smooth convex problems, the SCGD can be accelerated to converge at a rate of @math O(k-2 7) in the general case and @math O(k-4 5) in the strongly convex case. For nonconvex problems, we prove that any limit point generated by SCGD is a stationary point, for which we also provide the convergence rate analysis. Indeed, the stochastic setting where one wants to optimize compositions of expected-value functions is very common in practice. The proposed SCGD methods find wide applications in learning, estimation, dynamic programming, etc.