Introduction Algorithms & Interfaces Supports & Services Product Info Fact Sheet
Training Research MINLP Research
"An integrated field model combined with optimization presents many technological challenges in terms efficient algorithms to couple models, as well as models with optimization, and sufficient hardware capability to run the complex model." by Silvya Dewi Rahmawati in March 2012 PhD thesis.
Optimal Control Area :
Title : "Searching the optimal solution of the root nodes in Active Set Identification" - Active Set methods are good for warm starts, which are also better for solving mixed integer nonlinear optimization problems. The penalty/barrier parameters and trust region methods help to solve the active set identification problems. When one optimal solution of the root node is obtained, it is possible to use multi-start to try to find the global solution. If more root nodes are found, then multi-start with acceptable boundaries between nodes partitions and outer nodes partitions should be attempted for global optimization. It is based on the possibility of the bounds on the optimal value and of the feasible solutions found. Some hints on parallel implementation and hot starts of large scale problems with the powerful features of Knitro software are discussed. Explicit Euler, Implicit Euler and Trapezoidal methods will be examined.
Small and medium scale mixed integer nonlinear programming (MINLP) can be solved using a Branch and Bound variant called "spatial Branch and Bound" (sBB), where branching is allowed both on continuous and discrete variables that contribute to the gap between the original problem and its convex relaxation. For large scale variants it is good to use heuristics, such as Rounding, Feasibility Pump, Local Branching; or according to the problem structure to solve using special-purpose methods.
When the processes to be optimized are time-dependent, mixed-integer optimal control problem (MIOCP) is formally more general than MINLP, but after a certain way of discretization/decomposition a subclass of MINLP with specific features that need to be studied.
Useful methods are convexification, spatial branching, domain reduction, nonlinear cutting planes techniques and reformulation-linearization technique.
Definition : The mixed-integer optimal control problem, also called mixed-integer dynamic optimization (MIDO) problem, considers the computation of time dependent operating conditions (controls), discrete – binary or integer- decisions and time-independent parameters so as to minimize (or maximize) a performance index (or cost function) while keeping a set of constraints coming from safety and/or quality demands and environmental regulations.
Because of their non-convexity, an optimal solution is in general sought using Branch & Bound techniques (in the switched systems). These methods recursively partition/discretization the feasible set and obtain a lower bound on the optimal solution value by generating convex relaxation of the original problem. The introduction of Automatic Differentiation is important for Evaluating Derivatives to minimize the evaluation/rounding errors.
Reformulation : It is good to use an outer convexification with respect to the binary controls as suggested by Sebastian Sager. Sebastian Sager describes that the reformulated control problem has two main advantages compared to standard formulations or convexifications. First, especially for time-optimal control problems, the optimal solution of the relaxed problem will exhibit a bang-bang structure, and is thus already integer feasible. Second, theoretical results have recently been found that show that even for path-constrained and sensitivity-seeking arcs the optimal solution of the relaxed problem yields the exact lower bound on the minimum of the integer problem. This allows to calculate precise error estimates, if a coarser control discretization grid, a simplified switching structure for the optimization of switching times, or heuristics are used.
