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Aubin and H. Ben-Tal, Second-order and related extremality conditions in nonlinear programming , J. Theory Appl. Zentralblatt MATH: You have access to this content.
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The term "mathematical programming" is connected with the fact that the goal of solving various problems is choosing programs of action. The problems of linear, quadratic and convex programming have a common property: Local optimality implies global optimality.
The so-called multi-extremum problems , for which the indicated property does not hold, are both considerably more difficult and less investigated. Thus, the problem of convex programming reduces to the solution of a system of equations and inequalities. In mathematical programming one of the main directions concerns computational methods for solving extremum problems.follow
Optimality and Stability in Mathematical Programming
One of the widest used among these methods is the method of feasible directions. This problem is readily solved using, say, the standard simplex method. Minimization methods for non-smooth functions were successfully elaborated. One of the representatives of this class is the method of the generalized gradient.
Stochastic methods of minimization are also widely used. A characteristic feature of the computational aspect of the methods for solving the problems in mathematical programming is that the application of these methods is invariably connected with the utilization of electronic computers. The main reason for this is that the problems in mathematical programming that formalize situations of control of real systems involve a large amount of work which cannot be performed by manual computation.
Optimality conditions in mathematical programming and composite optimization
One of the widespread methods for investigating problems in mathematical programming is the method of penalty functions. This method essentially replaces the given mathematical programming problem by a sequence of parametric problems on unconditional minimization, such that when the parameter tends to infinity in other cases, to zero the solutions of these auxiliary problems converge to the solution of the original problem.
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An important direction of investigation in mathematical programming is the problem of stability. Here essential importance is attached to the study of the class of stable problems, that is, problems for which small perturbations errors in the data result in perturbations of the solutions that are also small.
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In the case of unstable problems an important role is reserved for a procedure of approximating the unstable problem by a sequence of stable problems — this is known as the regularization process. Along with finite-dimensional problems, problems of mathematical programming in infinite-dimensional spaces are considered as well. Among the latter there are various extremum problems in mathematical economics, technology, problems of optimization of physical characteristics of nuclear reactors, and others.
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In the terminology of mathematical programming one formulates the problems in the corresponding function space of variational calculus and optimal control. Mathematical programming has crystallized as a science in the s until s. This is first of all connected with the development of electronic computers and, hence, the mathematical processing of large amounts of data. On this basis one solves problems of control and planning, in which the application of mathematical methods is connected mainly with the construction of mathematical models and related extremal problems, among them problems of mathematical programming.
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