Sequential quadratic programming method

Fig. 6.7 Comparison of the maximum of the neural network approximation of the ODHE ethylene yield obtained in 10 runs of the genetic algorithm with a population size 60, and the global maximum obtained with a sequential quadratic programming method run for 15 different starting points.

Stoer, J. (1985), Principals of Sequential Quadratic Programming Methods for Solving Nonlinear Programs, in Computational Mathematical Programming, K. Schittkowski, Ed., Springer, Berlin. 

Solution methods for optimization problems that involve only continuous variables can be divided into two broad classes derivative-free methods (e.g., pattern search and stochastic search methods) and derivative-based methods (e.g., barrier function techniques and sequential quadratic programming). Because the optimization problems of concern in RTO are typically of reasonably large scale, must be solved on-line in relatively small amounts of time and derivative-free methods, and generally have much higher computational requirements than derivative-based methods, the solvers contained in most RTO systems use derivative-based techniques. Note that in these solvers the first derivatives are evaluated analytically and the second derivatives are approximated by various updating techniques (e.g., BFGS update). 

Problem Type Smooth nonlinear functions subject to smooth constraints Method Sequential quadratic programming Author Peter SpeUucci, Technical University Darmstadt, Germany Contact http //www.mathematik.tu-darmstadt.de/ags/ag8/spellucci/... 

Problem Type Multiple Unear/nonlinear objective functions with Unear/nonlinear constraints Method Sequential quadratic programming... 

Problem Type Large-scale linear and nonlinear programs Method Sparse sequential quadratic programming... 

The Lagrange function is approximated with a quadratic function, whereas the nonlinear constraints are linearized (Sequential Quadratic Programming, SQP, method see Section 13.7). Also in this case, a lower level BzzConstrai-nedMinimization class object with a quadratic objective function and hnear constraints is invoked. 

Optimization is performed solving a Multiobjective Optimization Problem (MOP) using as optimization algorithm a customized Sequential Quadratic Programming (SQP) method (Fletcher 1987). The solution of the MOP provides the Pareto front of the problem which, in our application, consists of 155 optimal solutions. However, as inputs of the equipment reliability and cost models fluctuate according to distribution laws reflecting uncertainty on parameters, objective functions will fluctuate also in repeated runs. 

Sequential quadratic programming (SQP) methods are iterative methods that solve at the Ml iteration a quadratic sub-problem (QP) of the form QP [22, 23] ... 

The MINLPsolver can be OAERAp based on outer proximation algorithms or SRQPD employs a sequential quadratic programming (SQP) method for the solution of the nonlinear progranuning (NLP) problem. 

Thus, the concentration of the chemical species has been determined in each of these regions. The numbers of phases and chemical species in each region have been chosen based on information reported in the literature. A Sequential Quadratic Programming (SQP) method has then been used to solve the minimisation problem. 

Pareto-optimal solutions, but it essentially works by solving a set of NLPs by means of the SQP (Sequential Quadratic Programming), which is a gradient-based method. Thus, it can fail with non-convex problems. In order to improve the robustness of the technique, we have replaced the SQP solver by SRES. 

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. 

One of its important applications is as the first phase in identifying which constraints are active and passive. For example, it is adopted in the SLQP method (Sequential Linear Quadratic Programming) (see Nocedal and Wright, 2000). 

The problem of minimum as formulated above can be solved by sequential methods of nonlinear (in particular quadratic) programming. The idea of the sequential approach consists, most simply, in linearizing the equation g(z) = 0 at point z of the sequence and subjecting the linearized constraint equation to a minimum condition thus the next approximation is found, and so on. Some problems can arise when the whole unmeasured vector y is not observable (not uniquely determined) although the latter case is less frequent in practice, possibly it can happen that the values of some unmeasured variables are not required and admitted as unobservable (undetermined). In what follows we shall outline two methods that do not require the full observability of vector y. 

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