Convex Mathematical Programming Problem Research Articles

Generalized Clarke epiderivatives of the extremum multifunction to a multi-objective parametric discrete optimal control problem

<p style='text-indent:20px;'>This paper deals with the generalized Clarke epiderivative of the extremum multifunction of a multi-objective parametric convex discrete optimal control problem with linear state equations and control constraints. By establishing an abstract result on the generalized epiderivative of the extremum multifunction of a multi-objective parametric convex mathematical programming problem, we derive a formula for computing the generalized Clarke epiderivative of the extremum multifunction to a multi-objective parametric convex discrete optimal control problem. Examples are given to illustrate the obtained results.</p>

Solution of the Two-Level Hierarchical Minimax Program Control Problem in a Nonlinear Discrete-Time Dynamical System

In this paper we consider a discrete-time dynamical system consisting of two controllable objects. The dynamics of each object is described by the corresponding nonlinear discrete-time vector recurrent equation. The quality of process implementation at each level of the control system is estimated by the corresponding terminal convex and differentiable functional. For the dynamical system under consideration, a mathematical formalization of a two-level hierarchical minimax program control problem in the presence of perturbations, and an algorithm for its solving are proposed. The construction of this algorithm can be implemented as a finite sequence of solutions of a linear and convex mathematical programming problems, and a finite discrete optimization problems.

Subgradients of the Value Function in a Parametric Convex Optimal Control Problem

Motivated by our recent works on the optimal value function in parametric optimal control problems under linear state equations, in this paper we study of the first-order behavior of the value function of a parametric convex optimal control problem with a convex cost function and linear state equations. By establishing an abstract result on the subdifferential of the value function to a parametric convex mathematical programming problem, we derive a formula for computing the subdifferential and the singular subdifferential of the value function to a parametric convex optimal control problem. By virtue of the convexity, several assumptions used in the above papers, like the existence of a local upper Lipschitzian selection of the solution map, as well as the V-inner semicontinuity of the solution map, are no longer needed.

Cloud Service for Interactive Simulation of Interregional Trade

The paper describes a mathematical model of trade flows between the territories of a region or a country in a transport network having one or more different types of marine or ground transportation. We use the approach of modeling complex communication systems to determine the most probable values of flows in case of incomplete information about the system. Transport costs between the territories are modeled within the framework of the gravity model. The payment for transportation depends on the distance between regions, the distance is estimated as the shortest way length in a given transport network or geographical distance. The mathematical formulation of the problem belongs to the class of convex mathematical programming problems and assumes the numerical solution of nonlinear optimization problem with linear constraints. Based on the model, the software is implemented as a cloud service on heterogeneous computing architectures: the simulation module is made on a highperformance server platform, management and visualization modules are produced with IACPaaS cloud platform. Communication between the platforms is established via asynchronous http-queries. For information exchange between the modules the declarative model with JSON format is developed and implemented for the objects considered in the mathematical model which are products, areas and communications. Visualization module allows to present graphically the original and the resulting matrix data and to modify the input parameters of the model interactively. The paper demonstrates the use of software for the simulation of inter-regional freight traffic of the Russian Far East region based on input data provided by open statistics sources.

Multiple criteria mathematical programming for multi-class classification and application in network intrusion detection

Multi-class classification problems are harder to solve and less studied than binary classification problems. The goal of this paper is to present a multi-criteria mathematical programming (MCMP) model for multi-class classification. Furthermore, we introduce the concept of e-support vector to facilitate computation of large-scale applications. Instead of finding the optimal solution for a convex mathematical programming problem, the computation of optimal solution for the model requires only matrix computation. Using two network intrusion datasets, we demonstrate that the proposed model can achieve both high classification accuracies and low false alarm rates for multi-class network intrusion classification.

Duality-based regularization in a linear convex mathematical programming problem

For a linear convex mathematical programming (MP) problem with equality and inequality constraints in a Hilbert space, a dual-type algorithm is constructed that is stable with respect to input data errors. In the algorithm, the dual of the original optimization problem is solved directly on the basis of Tikhonov regularization. It is shown that the necessary optimality conditions in the original MP problem are derived in a natural manner by using dual regularization in conjunction with the constructive generation of a minimizing sequence. An iterative regularization of the dual algorithm is considered. A stopping rule for the iteration process is presented in the case of a finite fixed error in the input data.

Sensitivity analysis and decomposition of unreliable production lines with blocking

The analysis of finite‐buffered, unreliable production lines is often based on the method of decomposition, where the original system is decomposed into a series of two‐stage subsystems that can be modeled as quasi birth‐death processes. In this paper, we present methods for computing the gradients of the equilibrium distribution vector for such processes. Then we consider a specific production line with finite buffers and machine breakdowns and develop an algorithm that incorporates gradient estimation into the framework of Gershwin's approximate decomposition. The algorithm is applied to the problem of workforce allocation to the machines of a production line to maximize throughput. It is shown that this problem is equivalent to a convex mathematical programming problem and, therefore, a globally optimal solution can be obtained.

Convex Duality and Generalized Solutions in Optimal Control Problem for Stopped Processes: The Deterministic Model

The approach consists of imbedding the original control problem tightly in a convex mathematical programming problem on the space of measures and then solving the latter problem by duality in convex analysis. The dual to the control problem is to find the supremum of all smooth subsolutions to the Bellman equation. Because of the effect of stops at the boundary of the domain, a different formulation of strong and weak problems will be adopted to make use of the convex duality method. The results about the decomposition of weak measures provide a clear intepretation for such a effect in the weak formualtion of our control problem.

On a Lagrangian penalty function method for nonlinear programming problems

We modify a Lagrangian penalty function method proposed in [4] for constrained convex mathematical programming problems in order to obtain a geometric rate of convergence. For nonconvex problems we show that a special case of the algorithm in the above paper is still convergent without coercivity and convexity assumptions.

Information theoretic multivariate graduation

Abstract This paper presents a new multivariate graduation method based upon a constrained information theoretic methodology. The statistical formulation of the graduation problem is shown to result in a convex mathematical programming problem thus allowing the actuary to include constraints such as monotonicity along the rows, monotonicity down the column, monotonicity along the upward diagonal, as well as convexity, and any of a variety of other relationships desired for the graduated series. An illustrative application is made to the graduation of select and ultimate mortality tables.

Convex Duality Approach to the Optimal Control of Diffusions

An alternative to the usual dynamic programming approach to the optimal control of Markov processes is considered. It is based on duality of convex analysis. The control problem is embedded in a convex mathematical programming problem on a space of measures. The dual problem is to find the supremum of smooth subsolutions to the Hamilton–Jacobi–Bellman equation.

On the convergence rate of the continuous version of the regularized gradient method

In this paper the convergence rate of the gradient type iterative regularization is given in the case when the convex mathematical programming problems additionally satisfy the condition of the normal regularity and the method is described in its continuous variant. The dependence of the convergence rate on the penalty and regularization parameters, on the stepsize and on the accuracity in the basic informations in formulated.

Optimal control of piecewise deterministic markov process

The trajectories of piecewise deterministic Markov processes are solutions of an ordinary (vector)differential equation with possible random jumps between the different integral curves. Both continuous deterministic motion and the random jumps of the processes are controlled in order to minimize the expected value of a performance functional consisting of continuous, jump and terminal costs. A limiting form of the Hamilton-Jacobi-Bellman partial differential equation is shown to be a necessary and sufficient optimality condition. The existence of an optimal strategy is proved and acharacterization of the value function as supremum of smooth subsolutions is also given. The approach consists of imbedding the original control problem tightly in a convex mathematical programming problem on the space of measures and then solving the latter by dualit

The Equivalence of Strong and Weak Formulations for Certain Problems in Optimal Control

It is shown that a wide class of control problems may be cast equivalently as convex mathematical programming problems over a space of Radon measures. The particularly simple structure of the equivalent problems will enable us to obtain new conditions for optimality, of dynamic programming type, by application of convex analysis. These implications are pursued in a companion paper.

Technical Note—Convex Programming with Set-Inclusive Constraints and Applications to Inexact Linear Programming

This note formulates a convex mathematical programming problem in which the usual definition of the feasible region is replaced by a significantly different strategy. Instead of specifying the feasible region by a set of convex inequalities, fi(x) ≦ bi, i = 1, 2, …, m, the feasible region is defined via set containment. Here n convex activity sets {Kj, j = 1, 2, …, n} and a convex resource set K are specified and the feasible region is given by [Formula: see text] where the binary operation + refers to addition of sets. The problem is then to find x̄ ∈ X that maximizes the linear function c · x. When the resource set has a special form, this problem is solved via an auxiliary linear-programming problem and application to inexact linear programming is possible.