We prove that the projected stochastic subgradient method, applied to a weakly convex problem, drives the gradient of the Moreau envelope to zero at the rateO(k−1/4).

In the modern digital economy, optimal decision support systems, as well machine learning are becoming an integral part of production processes. Artificial neural network training other engineering problems generate such high dimension that difficult to solve with traditional gradient or conjugate methods. Relaxation subgradient minimization methods (RSMMs) construct a descent direction forms o...

The Traveling Salesman Problem (TSP) is a classical Combinatorial Optimization problem, which has been intensively studied. The Lagrangean relaxation was first applied to the TSP in 1970. The Lagrangean relaxation limit approximates what is known today as HK (Held and Karp) bound, a very good bound (less than 1% from optimal) for a large class of symmetric instances. It became a reference bound...

The paper provides two contributions. First, we present new convergence results for conditional e-subgradient algorithms for general convex programs. The results obtained here extend the classical ones by Polyak [Sov. Math. Doklady 8 (1967) 593; USSR Comput. Math. Math. Phys. 9 (1969) 14; Introduction to Optimization, Optimization Software, New York, 1987] as well as the recent ones in [Math. P...