Algebraic inequality constraints optimization

The ingredients of formulating optimization problems include a mathematical model of the system, an objective function that quantifies a criterion to be extremized, variables that can serve as decisions, and, optionally, inequality constraints on the system. When represented in algebraic form, the general formulation of discrete/continu-ous optimization problems can be written as the following mixed integer optimization problem ... 

All of the various optimization techniques described in previous chapters can be applied to one or more types of reactor models. The reactor model forms a set of constraints so that most optimization problems involving reactors must accommodate steady-state algebraic equations or dynamic differential equations as well as inequality constraints. 

Algebraic optimization with equality and inequality constraints... 

In summary, condition 1 gives a set of n algebraic equations, and conditions 2 and 3 give a set of m constraint equations. The inequality constraints are converted to equalities using h slack variables. A total of M + m constraint equations are solved for n variables and m Lagrange multipliers that must satisfy the constraint qualification. Condition 4 determines the value of the h slack variables. This theorem gives an indirect problem in which a set of algebraic equations is solved for the optimum of a constrained optimization problem. 

As mentioned earlier, the developed algorithm employs dynopt to solve the intermediate problems associated with the local interaction of the agents. Specifically, dynopt is a set of MATLAB functions that use the orthogonal collocation on finite elements method for the determination of optimal control trajectories. As inputs, this toolbox requires the dynamic process model, the objective function to be minimized, and the set of equality and inequality constraints. The dynamic model here is described by the set of ordinary differential equations and differential algebraic equations that represent the fermentation process model. For the purpose of optimization, the MATLAB Optimization Toolbox, particularly the constrained nonlinear rninimization routine fmincon [29], is employed. 

Based on a multiple shooting method for parameter identification in differential-algebraic equations due to Heim [4], a new implementation of a direct multiple shooting method for optimal control problems has been developed, which enables the solution of problems that can be separated into different phases. In each of these phases, which might be of unknown length, the control behavior due to inequality constraints, the differential equations, even the dimensions of the state and/or the control space can differ. For the optimal control problems under investigation, the different phases are concerned with the different steps of the recipes. 

---