arXiv subjects

Roman Polyak

Publications and source records attributed to Roman Polyak.

4 recordsLinked to original sources

arXiv2022

Finding Nonlinear Production -- Consumption Equilibrium

We introduce and study nonlinear production - consumption equilibrium (NPCE). The NPCE is a combination and generalization of both classical linear programming (LP) and classical input-output (IO) models. In contrast to LP and IO the NPCE has both production and consumption components. Moreover, the production cost, the consumption and the factors (resources) availability are not fixed. Instead they are corespondent functions of the production output, prices of goods and prices of factors. At the NPCE the total production cost reaches its minimum, while the total consumption, without factors expences, reaches its maximum. At the same time the production cost is consistent with the production output, the consumption is consistent with the prices for goods and the factors availability is consistent with prices for factors. Finding NPCE is equivalent to solving a variational inequality (VI) with a particular nonlinear operator and a simple feasible set. Under natural assumptions on the production, consumption and factor operators the NPCE exists and it is unique. Projection on the feasible set {\Omega} is a low cost operation, therefore for solving the VI we use two projection methods. Each of them requires at each step few matrix by vector multiplications and allows, along with convergence and convergence rate, establish complexity bound. The methods decompose the problem, so both the primal and the dual variables are computed simultaneously. On the other hand, both methods are pricing mechanisms for establishing NPCE, which is a generalization of Walras - Wald equilibrium (see [12]) in few directions.

math.OC↗

arXiv2017

Complexity of the Regularized Newton Method

Newton's method for finding an unconstrained minimizer for strictly convex functions, generally speaking, does not converge from any starting point. We introduce and study the damped regularized Newton's method (DRNM). It converges globally for any strictly convex function, which has a minimizer in $R^n$. Locally DRNM converges with a quadratic rate. We characterize the neighborhood of the minimizer, where the quadratic rate occurs. Based on it we estimate the number of DRNM's steps required for finding an $\varepsilon$- approximation for the minimizer.

math.OC↗

arXiv2017

Exterior Distance Function

We introduce and study exterior distance function (EDF) and correspondent exterior point method (EPM) for convex optimization. The EDF is a classical Lagrangian for an equivalent problem obtained from the initial one by monotone transformation of both the objective function and the constraints. The constraints transformation is scaled by a positive scaling parameter. Thus, the EDF is a particular realization of the Nonlinear Rescaling (NR) principle. Along with the "center", the EDF has two extra tools: the barrier (scaling) parameter and the vector of Lagrange multipliers. We show that EPM generates primal - dual sequence, which converges to the primal - dual solution in value under minimum assumption on the input data. Moreover, the convergence is taking place under any fixed interior point as a "center" and any fixed positive scaling parameter, just due to the Lagrange multipliers update. If the second order sufficient optimality condition is satisfied, then the EPM converges with Q-linear rate under any fixed interior point as a "center" and any fixed, but large enough positive scaling parameter.

math.OC↗

arXiv2016

The Legendre Transform in Modern Optimization

The Legendre transform (LET) is a product of a general duality principle: any smooth curve is, on the one hand, a locus of pairs, which satisfy the given equation and, on the other hand, an envelope of a family of its tangent lines. An application of the LET to a strictly convex and smooth function leads to the Legendre identity (LEID). For strictly convex and three times differentiable function the LET leads to the Legendre invariant (LEINV). Although the LET has been known for more then 200 years both the LEID and the LEINV are critical in modern optimization theory and methods. The purpose of the paper (survey) is to show the role of the LEID and the LEINV play in both constrained and unconstrained optimization.

math.OC↗