Constrained Optimization - Dependent Variables

An important class of the constrained optimization problems is one in which the objective function, equality constraints and inequality constraints are all linear. A linear function is one in which the dependent variables appear only to the first power. For example, a linear function of two variables x and x2 would be of the general form ... 

When solving an inequality-constrained optimal control problem numerically, it is impossible to determine which constraints are active. The reason is one cannot obtain a p, exactly equal to zero. This difficulty is surmounted by considering a constraint to be active if the corresponding p < a where a is a small positive number such as 10 or less, depending on the problem. Slack variables may be used to convert inequalities into equalities and utilize the Lagrange Multiplier Rule. 

The question of constrained optimisation is answered in a standard manner by Euler-Lagrange optimisations. By formulating the problem with optimal control theory, Johannessen and Kjelstrup explained that the Hamiltonian of the problem was constant in a study of chemical reactors. The total entropy production for a plug flow reactor was written as a function of a position-dependent state variable vector x(z) and the control variable u(z) ... 

The efficient and accurate solution to the optimal problem is not only dependent on the size of the problem in terms of the number of constraints and design variables but also on the characteristics of the objective function and constraints. When both the objective function and the constraints are linear functions of the design variable, the problem is known as a LP problem. Quadratic programming (QP) concerns the minimization or maximization of a quadratic objective function that is linearly constrained. For both the LP and QP problems, reliable solution procedures are readily available. More difficult to solve is the NLP problem in which the objective function and constraints may be nonlinear functions of the design variables. A solution of the NLP problem generally requires an iterative procedure to establish a direction of search at each major iteration. This is usually achieved by the solution of an LP, a QP, or an unconstrained subproblem. 

The methods for solving an optimization task depend on the problem classification. Since the maximum of a function / is the minimum of the function —/, it suffices to deal with minimization. The optimization problem is classified according to the type of independent variables involved (real, integer, mixed), the number of variables (one, few, many), the functional characteristics (linear, least squares, nonlinear, nondifferentiable, separable, etc.), and the problem. statement (unconstrained, subject to equality constraints, subject to simple bounds, linearly constrained, nonlinearly constrained, etc.). For each category, suitable algorithms exist that exploit the problem s structure and formulation. 

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