Constrained optimization quadratic programming

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 application of the standard nonlinear programming techniques of constrained optimization on analyzing the mean and variance response surfaces has been investigated by Del Castillo and Montgomery [34]. These techniques are appropriate since both the primary and secondary responses are usually quadratic functions. 

P. M. Pardalos and G. Schnitger. Checking local optimality in constrained quadratic programming is NP-Hard. Oper. Res. Lett., 7(1) 33,1988. 

In addition to a wide variety of problem types, there are three common types of constrained optimization problems that are typically of interest linear programs (LPs), quadratic programs (QPs), and nonlinear programs (NLPs). 

There are essentially six types of procedures to solve constrained nonlinear optimization problems. The three methods considered more successful are the successive LP, the successive quadratic programming, and the generalized reduced gradient method. These methods use different strategies but the same information to move from a starting point to the optimum, the first partial derivatives of the economic model, and constraints evaluated at the current point. Successive LP is used in a number of solvers including MINOS. Successive quadratic programming is the method of... 

Q, R) model, 1671 QC, see Quality circle QCD (quality, cost, and delivery), 552 QFD, see Quality function deployment QFP (quad flat pack), 424 Q1 teams, see Quality improvement teams Q-leaming, 1780 QLs (query languages), 119 QMS standards, see Quality management systems standards QoS, see Quality of service QR system, see Quick response system QS 9000 standard, 1973 Quad flat pack (QFP), 424 Quadratic programming problems (constrained optimization), 2555, 2562 Quality. See also Rehabihty in advanced planning and scheduling (APS), 2049... 

Another important application of this possibility of transforming a constrained optimization problem into an equivalent (dual) problem is in quadratic programming (Chapter 11), where the objective function is a quadric and all constraints are ... 

A great number of studies indicated that quadratic approximation methods, which are characterized by solving a sequence of quadratic subproblems recursively belong to the most efficient and reliable nonlinear programming algorithms presently available. This method combines the most efficient characteristics of different optimization techniques (see e.g, [19]). For equality constrained problems, the general nonlinear constrained optimization problem can be formulated by an... 

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. 

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. 

The introduction of inequality constraints results in a constrained optimization problem that can be solved numerically using linear or quadratic programming techniques (Edgar et al., 2001). As an example, consider the addition of inequality constraints to the MFC design problem in the previous section. Suppose that it is desired to calculate the M-step control policy AU(k) that minimizes the quadratic objective function J in Eq. 20-54, while satisfying the constraints in Eqs. 20-59, 20-60, and 20-61. The output predictions are made using the step-response model in Eq. 20-36. This MFC... 

The augmented Lagrangian method is not the only approach to solving constrained optimization problems, yet a complete discussion of this subject is beyond the scope of this text. We briefly consider a popular, and efficient, class of methods, as it is used by fmincon, sequential quadratic programming (SQP). We wUl find it useful to introduce a common notation for the equality and inequality constraints using slack variables. 

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