Constrained, multivariate optimization

Jang, S. S., Joseph, B., and Mukai, H., On-line optimization of constrained multivariable... 

Prett, D.M., Gillette, R.D. Optimization and Constrained Multivariable Control of a Catalytic Cracking Unit (1979) AIChE 1979 National Meeting, Houston... 

The optimality conditions discussed in the previous sections formed the theoretical basis for the development of several algorithms for unconstrained and constrained nonlinear optimization problems. In this section, we will provide a brief outline of the different classes of nonlinear multivariable optimization algorithms. 

Closed-loop multivariable boiler control has to be planned and performed carefully because plant operators are not traditionally willing to reduce air-fuel ratios due to concerns about CO and other symptoms associated with Oz-deficient combustion. Model predictive control (MPC) is by far the most widely used technique for conducting multivariable boiler optimization and control. Forms of MPC that are inherently multivariable and that include real-time constrained optimization in the design are best suited for boiler application. 

In the multivariable case with differently effective inputs, the estimation of the appropriate bandwidth is difficult in general. One may resort to a simulation of a time-optimal controller or a constrained LQ-MPC controller to estimate the attainable rise times. 

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. 

Geometry optimization problems for molecules in the context of standard all-atom force fields in computational chemistry are typically of the multivariate, continuous, and nonlinear type. They can be formulated as constrained (as in adiabatic relaxation) or unconstrained. Discontinuities in the derivatives may be a problem in certain formulations involving truncation, such as of the nonbonded terms (see Section 7). 

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