Nonlinear programming problem Unconstrained optimization

In nonlinear programming problems, optimal solutions need not occur at vertices and can occur at points with positive degrees of freedom. It is possible to have no active constraints at a solution, for example in unconstrained problems. We consider nonlinear problems with constraints in Chapter 8. 

Constrained Optimization When constraints exist and cannot be eliminated in an optimization problem, more general methods must be employed than those described above, because the unconstrained optimum may correspond to unrealistic values of the operating variables. The general form of a nonlinear programming problem allows for a nonlinear objective function and nonlinear constraints, or... 

Nonlinear Programming The most general case for optimization occurs when both the objective function and the constraints are nonlinear, a case referred to as nonlinear programming. While the ideas behind the search methods used for unconstrained multivariable problems are applicable, the presence of constraints complicates the solution procedure. All the methods discussed below have been utilized to solve nonlinear programming problems in the field of chemical engineering design and operations. Nonlinear programming is now used extensively in the area of real-time optimization. 

Most practical multivariable problems include constraints, which must be treated using enhancements of unconstrained optimization algorithms. The next two sections describe two classes of constrained optimization techniques that are used extensively in the process industries. When constraints are an important part of an optimization problem, constrained techniques must be employed, because an unconstrained method might produce an optimum that violates the constraints, leading to unrealistic values of the process variables. The general form of an optimization problem includes a nonlinear objective function (profit) and nonlinear constraints and is called a nonlinear programming problem. 

Problem 4.1 is nonlinear if one or more of the functions/, gv...,gm are nonlinear. It is unconstrained if there are no constraint functions g, and no bounds on the jc,., and it is bound-constrained if only the xt are bounded. In linearly constrained problems all constraint functions g, are linear, and the objective/is nonlinear. There are special NLP algorithms and software for unconstrained and bound-constrained problems, and we describe these in Chapters 6 and 8. Methods and software for solving constrained NLPs use many ideas from the unconstrained case. Most modem software can handle nonlinear constraints, and is especially efficient on linearly constrained problems. A linearly constrained problem with a quadratic objective is called a quadratic program (QP). Special methods exist for solving QPs, and these iare often faster than general purpose optimization procedures. 

The simplex mefliod, implemented as fminsearch in the optional MATLAB optimization tool kit, requires only aroutine thatretums F x). WWle simplex methods are used commonly for linear programming problems with linear cost functions and constraints (Nocedal Wright, 1999), for unconstrained optimization with nonlinear cost functions, the gradient and Newton methods discussed below are preferred. Thus, we provide here only a cursory description, and refer the interested reader to the supplemental material in the accompanying website for further details. 

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. 

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