A minimization algorithm with approximation of an epigraph of the objective function and a constraint set

Copyright © by the paper's authors.An algorithm is suggested for solving a convex programming problem which belongs to a class of cutting methods. In the algorithm an epigraph of the objective function and a feasible solutions set of the problem are embedded into some auxiliary sets to construct iteration points. Since these embedded sets are constructed as polyhedral sets in the algorithm, then each iteration point is found by solving a linear programming problem independently of the type of functions which define the initial problem. The suggested algorithm is characterized by the following fact. Sets which approximate the epigraph of the objective function can be updated periodically on the base of discarding cutting planes

Variant of the cutting plane method with approximation of the set of constraints and auxiliary functions epigraphs

© Springer International Publishing AG, part of Springer Nature 2018. We propose a method of solving a convex programming problem, which is based on the ideas of cutting plane methods and the method of penalty functions. To construct each approximation, the method uses an operation of immersing the epigraph of auxiliary function into a polyhedral set. The auxiliary function is constructed as the sum of the objective function and the external penalty function of the constraint area. In addition, an admissible set of the original problem is immersed in the polyhedron simultaneously. In connection with this, the problem of constructing an iterative point is a linear programming problem, in which constraints are polyhedrons approximating the epigraph of auxiliary function and the admissible set. Both next approximating sets are based on the previous ones by cutting off the iterative point from them by hyperplanes. The convergence of the method is proved. We describe its algorithms. One of them can be the implementation of the method of penalty functions

A minimization algorithm with approximation of an epigraph of the objective function and a constraint set

Copyright © by the paper's authors.An algorithm is suggested for solving a convex programming problem which belongs to a class of cutting methods. In the algorithm an epigraph of the objective function and a feasible solutions set of the problem are embedded into some auxiliary sets to construct iteration points. Since these embedded sets are constructed as polyhedral sets in the algorithm, then each iteration point is found by solving a linear programming problem independently of the type of functions which define the initial problem. The suggested algorithm is characterized by the following fact. Sets which approximate the epigraph of the objective function can be updated periodically on the base of discarding cutting planes

A minimization algorithm with approximation of an epigraph of the objective function and a constraint set

Copyright © by the paper's authors.An algorithm is suggested for solving a convex programming problem which belongs to a class of cutting methods. In the algorithm an epigraph of the objective function and a feasible solutions set of the problem are embedded into some auxiliary sets to construct iteration points. Since these embedded sets are constructed as polyhedral sets in the algorithm, then each iteration point is found by solving a linear programming problem independently of the type of functions which define the initial problem. The suggested algorithm is characterized by the following fact. Sets which approximate the epigraph of the objective function can be updated periodically on the base of discarding cutting planes