Nonlinear programming problem convex

In summary, the optimum of a nonlinear programming problem is, in general, not at an extreme point of the feasible region and may not even be on the boundary. Also, the problem may have local optima distinct from the global optimum. These properties are direct consequences of nonlinearity. A class of nonlinear problems can be defined, however, that are guaranteed to be free of distinct local optima. They are called convex programming problems and are considered in the following section. 

An important result in mathematical programming evolves from the concept of convexity. For the nonlinear programming problem called the convex programming problem... 

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. 

The sufficient conditions for obtaining a global solution of the nonlinear programming problem are that both the objective function and the constraint set be convex. If these conditions are not satisfied, there is no guarantee that the local optima will be the global optima. 

The next iteration consists of creating the new convex hull of concentrations, which include the concentrations obtained by this extension, and checking for favorable recycle reactor extensions from this new convex hull. At iteration P, this involves the solution of the following nonlinear programming problem ... 

A quadratic programming problem minimizes a quadratic function of n variables subject to m linear inequality or equality constraints. A convex QP is the simplest form of a nonlinear programming problem with inequality constraints. A number of practical optimization problems are naturally posed as a QP problem, such as constrained least squares and some model predictive control problems. 

As shown in Fig. 3-53, optimization problems that arise in chemical engineering can be classified in terms of continuous and discrete variables. For the former, nonlinear programming (NLP) problems form the most general case, and widely applied specializations include linear programming (LP) and quadratic programming (QP). An important distinction for NLP is whether the optimization problem is convex or nonconvex. The latter NLP problem may have multiple local optima, and an important question is whether a global solution is required for the NLP. Another important distinction is whether the problem is assumed to be differentiable or not. 

Although convexity is desirable, many real-world problems turn out to be non-convex. In addition, there is no simple way to demonstrate that a nonlinear problem is a convex problem for all feasible points. Why, then is convex programming studied The main reasons are... 

Branch and bound (BB) is a class of methods for linear and nonlinear mixed-integer programming. If carried to completion, it is guaranteed to find an optimal solution to linear and convex nonlinear problems. It is the most popular approach and is currently used in virtually all commercial MILP software (see Chapter 7). 

Konno and Yamazaki (1991) proposed a large-scale portfolio optimization model based on mean-absolute deviation (MAD). This serves as an alternative measure of risk to the standard Markowitz s MV approach, which models risk by the variance of the rate of return of a portfolio, leading to a nonlinear convex quadratic programming (QP) problem. Although both measures are almost equivalent from a mathematical point-of-view, they are substantially different computationally in a few perspectives, as highlighted by Konno and Wijayanayake (2002) and Konno and Koshizuka (2005). In practice, MAD is used due to its computationally-attractive linear property. 

This paper presents a general mathematical programming formulation the can be used to obtain customized tuning for PID controllers. A reformulation of the initial NLP problem is presented that transforms the nonlinear formulation to a linear one. In the cases where the objective function is convex then the resulting formulation can be solved easily to global optimality. The usefulness of the proposed formulation is demonstrated in five case studies where some of the most commonly used models in the process industry are employed. It was shown that the proposed methodology offers closed loop performance that is comparable to the one... 

With SLP there is no guarantee of convergence or global optimality. The method is widely used, nonetheless, as it is a simple extension of linear programming. It should be noted that whenever discontinuous linear functions are used to approximate a nonlinear function, then the problem behaves like an SLR There is no guarantee of convexity or convergence to the optimal solution. 

Nonlinear problems— NLP and MINLP—can be further classified as convex or nonconvex, depending on the convexity of the objective function and feasible region. The understanding of the type of problem in terms of classification and convexity is very important in the utilization of modeling systems, since there are specific solvers and solution techniques for each type of problem, and depending on the problem, there may be local and global solutions. A more comprehensive study on mathanati-cal programming topics and continuous nonlinear optimization is out of the scope of this chapter. The interested reader is directed to references [4,6,7]. 

---