Nonlinear programming problem example

Another method for solving nonlinear programming problems is based on quadratic programming (QP)1. Quadratic programming is an optimization procedure that minimizes a quadratic objective function subject to linear inequality or equality (or both types) of constraints. For example, a quadratic function of two variables x and X2 would be of the general form ... 

Optimization of a distributed parameter system can be posed in various ways. An example is a packed, tubular reactor with radial diffusion. Assume a single reversible reaction takes place. To set up the problem as a nonlinear programming problem, write the appropriate balances (constraints) including initial and boundary conditions using the following notation ... 

For example, it is usually impossible to prove that a given algorithm will find the global minimum of a nonlinear programming problem unless the problem is convex. For nonconvex problems, however, many such algorithms find at least a local minimum. Convexity thus plays a role much like that of linearity in the study of dynamic systems. For example, many results derived from linear theory are used in the design of nonlinear control systems. 

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. 

Unlike linear programming, in which an optimum, if one exists, can be found at an extreme point of the feasible region, solutions of nonlinear programming problems can occur at any feasible point. Whereas Figure 1 shows an example where the optimum lies on the boundary of the feasible region. Figure 2 illustrates a case where the optimum is an interior point of the feasible region. In the latter... 

In an earlier section, we had alluded to the need to stop the reasoning process at some point. The operationality criterion is the formal statement of that need. In most problems we have some understanding of what properties are easy to determine. For example, a property such as the processing time of a batch is normally given to us and hence is determined by a simple database lookup. The optimal solution to a nonlinear program, on the other hand, is not a simple property, and hence we might look for a simpler explanation of why two solutions have equal objective function values. In the case of our branch-and-bound problem, the operationality criterion imposes two requirements ... 

Neither of the problems illustrated in Figures 4.5 and 4.6 had more than one optimum. It is easy, however, to construct nonlinear programs in which local optima occur. For example, if the objective function / had two minima and at least one was interior to the feasible region, then the constrained problem would have two local minima. Contours of such a function are shown in Figure 4.7. Note that the minimum at the boundary point x1 = 3, x2 = 2 is the global minimum at / = 3 the feasible local minimum in the interior of the constraints is at / = 4. 

Using this nonlinear programming approach (also termed the embedded model or feasible path approach), we denote as x the vector of parameters representing l/(t) as well as the parameters x. For example, if U t) is assumed piecewise constant over a variable distance, we include w, and t in x. Problem... 

Note that independent variable, time, disappears from this problem. While final time constraints in (16) appear naturally in (17), other constraints that need to be enforced over the time domain are difficult to handle. For example, Sargent and Sullivan (1977) converted these to final time constraints by integrating the square of the constraint violations and forcing these to be less than a tolerance at final time this however, leads to degeneracies in solving the nonlinear program. 

Linear and nonlinear programming solvers have been interfaced to spreadsheet software for desktop computers. The spreadsheet has become a popular user interface for entering and manipulating numeric data. Spreadsheet software increasingly incorporates analytic tools that are accessible from the spreadsheet interface and permit access to external databases. For example, Microsoft Excel incorporates an optimization-based routine called Solver that operates on the values and formulas of a spreadsheet model. Current versions (4.0 and later) include LP and NLP solvers and mixed integer programming (MIP) capability for both linear and nonlinear problems. The user specifies a set of cell addresses to be independently adjusted (the decision variables), a set of formula cells whose values are to be constrained (the constraints), and a formula cell designated as the optimization objective. 

This approach enables the incorporation of input, output, and final-time constraints and is flexible with respect to the crystallizer configuration, the objective function definition, and the choice of manipulated variables. Examples of the use of nonlinear programming to solve this problem are given subsequently. 

If all of the functions /, g, hj are linear functions of x, then (P) is called a linear program. Otherwise (P) is a nonlinear program. Note that Example 1 is a nonlinear program since the objective function (1) is a nonlinear function. Actually, as will be seen later, this problem can be classified as a quadratic program since the objective function is a quadratic function and the constraints are all linear functions. 

An important aspect in LP is the requirement that all mathematical expressimis must be Unear. Obviously, in practice, not all factors are linear, so nonlinear expressions are sometimes necessary to model specific situations, for example, in the case of economies of scale. Fortunately, it has been shown that many factors (work hours, use of machinery, benefits) are reasonably linear or can be approximated by expressions of this type. However, if there is found to be a factor that definitely cannot be expressed linearly, then the problem caimot be solved by LP. This is certainly is the case with nonlinear programming (NLP), which is beyond the scope of this textbook. 

Nonlinear IP problems can be converted to linear integer programs using binary variables. We shall illustrate this with a numerical example. 

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