A fast adaptive heuristic stochastic gradient descent method is proposed. It is shown that this algorithm has a higher convergence rate in practical problems than currently popular optimization methods. Furthermore, a justification of this method is given, and difficulties that prevent obtaining optimal estimates for the proposed algorithm are described.
A strongly convex function of simple structure (for example, separable) is minimized under affine constraints. A dual problem is constructed and solved by applying a fast gradient method. The necessary properties of this method are established relying on which, under rather general conditions, the solution of the primal problem can be recovered with the same accuracy as the dual solution from the sequence generated by this method in the dual space of the problem. Although this approach seems natural, some previously unpublished rather subtle results necessary for its rigorous and complete theoretical substantiation in the required generality are presented
The problem of equilibrium distribution of flows in a transportation network in which a part of edges are characterized by cost functions and the other edges are characterized by their capacity and constant cost for passing through them if there is no congestion is studied. Such models (called mixed models) arise, e.g., in the description of railway cargo transportation. A special case of the mixed model is the family of equilibrium distribution of flows over routes—BMW (Beckmann) model and stable dynamics model. The search for equilibrium in the mixed model is reduced to solving a convex optimization problem. In this paper, the dual problem is constructed that is solved using the mirror descent (dual averaging) algorithm. Two different methods for recovering the solution of the original (primal) problem are described. It is shown that the proposed approaches admit efficient parallelization. The results on the convergence rate of the proposed numerical methods are in agreement with the known lower oracle bounds for this class of problems (up to multiplicative constants).
Previous and new results are used to compare two mathematical insurance models with identical insurance company strategies in a financial market, namely, when the entire current surplus or its constant fraction is invested in risky assets (stocks), while the rest of the surplus is invested in a risk-free asset (bank account). Model I is the classical Cramér–Lundberg risk model with an exponential claim size distribution. Model II is a modification of the classical risk model (risk process with stochastic premiums) with exponential distributions of claim and premium sizes. For the survival probability of an insurance company over infinite time (as a function of its initial surplus), there arise singular problems for second-order linear integrodifferential equations (IDEs) defined on a semiinfinite interval and having nonintegrable singularities at zero: model I leads to a singular constrained initial value problem for an IDE with a Volterra integral operator, while II model leads to a more complicated nonlocal constrained problem for an IDE with a non-Volterra integral operator. A brief overview of previous results for these two problems depending on several positive parameters is given, and new results are presented. Additional results are concerned with the formulation, analysis, and numerical study of “degenerate” problems for both models, i.e., problems in which some of the IDE parameters vanish; moreover, passages to the limit with respect to the parameters through which we proceed from the original problems to the degenerate ones are singular for small and/or large argument values. Such problems are of mathematical and practical interest in themselves. Along with insurance models without investment, they describe the case of surplus completely invested in risk-free assets, as well as some noninsurance models of surplus dynamics, for example, charity-type models.
The one-dimensional quasi-gasdynamic system of equations in the form of mass, momentum, and total energy conservation laws with general gas equations of state is considered. A family of three-point symmetric spatial discretizations of this system is studied for which the internal energy equation has a suitable form (without imbalance terms). An entropy balance equation is derived, and the influence exerted by the choice of discretizations of various terms on the form of difference imbalance terms in this equation is determined. Special discretizations are presented for which the corresponding nondivergence imbalance terms are zero. The Euler system of equations is solved numerically in the cases of a perfect polytropic gas, stiffened gas, and the van der Waals equations of state.