The projective exact penalty method for general constrained optimization

A new projective exact penalty function method is proposed for the equivalent reduction of constrained optimization problems to nonsmooth unconstrained ones. In the method, the original objective function is extended to infeasible points by summing its value at the projection of an infeasible point on the feasible set with the distance to the projection. The … Read more

Stochastic Optimization Models of Insurance Mathematics

The paper overviews stochastic optimization models of insurance mathematics and methods for their solution from the point of view of stochastic programming and stochastic optimal control methodology, with vector optimality criteria. The evolution of an insurance company‚Äôs capital is considered in discrete time. The main random variables, which influence this evolution, are levels of payments, … Read more

Stochastic generalized gradient methods for training nonconvex nonsmooth neural networks

The paper observes a similarity between the stochastic optimal control of discrete dynamical systems and the learning multilayer neural networks. It focuses on contemporary deep networks with nonconvex nonsmooth loss and activation functions. The machine learning problems are treated as nonconvex nonsmooth stochastic optimization problems. As a model of nonsmooth nonconvex dependences, the so-called generalized … Read more

Substantiation of the Backpropagation Technique via the Hamilton-Pontryagin Formalism for Training Nonconvex Nonsmooth Neural Networks

The paper observes the similarity between the stochastic optimal control of discrete dynamical systems and the training multilayer neural networks. It focuses on contemporary deep networks with nonconvex nonsmooth loss and activation functions. In the paper, the machine learning problems are treated as nonconvex nonsmooth stochastic optimization problems. As a model of nonsmooth nonconvex dependences, … Read more

Generalized Gradients in Problems of Dynamic Optimization, Optimal Control, and Machine Learning

In this work, nonconvex nonsmooth problems of dynamic optimization, optimal control in discrete time (including feedback control), and machine learning are considered from a common point of view. An analogy is observed between tasks of controlling discrete dynamic systems and training multilayer neural networks with nonsmooth target function and connections. Methods for calculating generalized gradients … Read more

Sample Average Approximation Method for Compound Stochastic Optimization Problems

The paper studies stochastic optimization (programming) problems with compound functions containing expectations and extreme values of other random functions as arguments. Compound functions arise in various applications. A typical example is a variance function of nonlinear outcomes. Other examples include stochastic minimax problems, econometric models with latent variables, and multilevel and multicriteria stochastic optimization problems. … Read more

REDUCTION OF TWO-STAGE PROBABILISTIC OPTIMIZATION PROBLEMS WITH DISCRETE DISTRIBUTION OF RANDOM DATA TO MIXED INTEGER PROGRAMMING PROBLEMS

We consider models of two-stage stochastic programming with a quantile second stage criterion and optimization models with a chance constraint on the second stage objective function values. Such models allow to formalize requirements to reliability and safety of the system under consideration, and to optimize the system in extreme conditions. We suggest a method of … Read more

On reducing a quantile optimization problem with discrete distribution to a mixed integer programming problem

We suggest a method for equivalent transformation of a quantile optimization problem with discrete distribution of random parameters to mixed integer programming problems. The number of additional integer (in fact boolean) variables in the equivalent problems equals to the number of possible scenarios for random data. The obtained mixed integer problems are solved by standard … Read more

On the Safety First portfolio selection

A.D.Roy’s (1952) safety first (SF) approach to a financial portfolio selection is improved. Safety first means minimization of probability of poor returns. Improvement concerns a better estimation of the poor return probabilities by means of shortfall risk functions. Optimal SF-portfolio is sought similar to Roy’s geometric method but with a different efficient frontier. In case … Read more

On mixed integer reformulations of monotonic probabilistic programming problems with discrete distributions

The paper studies large scale mixed integer reformulation approach to stochastic programming problems containing probability and quantile functions, under assumption of discreteness of the probability distribution involved. Jointly with general sample approximation technique and contemporary mixed integer programming solvers the approach gives a regular framework to solution of practical probabilistic programming problems. In the literature … Read more