Optimal solution strategy

If the matrix Q is positive semidefinite (positive definite) when projected into the null space of the active constraints, then (3-98) is (strictly) convex and the QP is a global (and unique) minimum. Otherwise, local solutions exist for (3-98), and more extensive global optimization methods are needed to obtain the global solution. Like LPs, convex QPs can be solved in a finite number of steps. However, as seen in Fig. 3-57, these optimal solutions can lie on a vertex, on a constraint boundary, or in the interior. A number of active set strategies have been created that solve the KKT conditions of the QP and incorporate efficient updates of active constraints. Popular methods include null space algorithms, range space methods, and Schur complement methods. As with LPs, QP problems can also be solved with interior point methods [see Wright (1996)]. 

A broad class of optimization strategies does not require derivative information. These methods have the advantage of easy implementation and little prior knowledge of the optimization problem. In particular, such methods are well suited for quick and dirty optimization studies that explore the scope of optimization for new problems, prior to investing effort for more sophisticated modeling and solution strategies. Most of these methods are derived from heuristics that naturally spawn numerous variations. As a result, a very broad literature describes these methods. Here we discuss only a few important trends in this area. 

With feasible path strategies, as the name implies, on each iteration you satisfy the equality and inequality constraints. The results of each iteration, therefore, provide a candidate design or feasible set of operating conditions for the plant, that is, sub-optimal. Infeasible path strategies, on the other hand, do not require exact solution of the constraints on each iteration. Thus, if an infeasible path method fails, the solution at termination may be of little value. Only at the optimal solution will you satisfy the constraints. 

This problem can be solved using a combined optimization and constraint model solution strategy (Muske and Edgar, 1998) by converting the differential equations to algebraic constraints using orthogonal collocation or some other model discretization approach. 

The classic methods use an ODE solver in combination with an optimization algorithm and solve the problem sequentially. This solution strategy is referred to as a sequential solution and optimization approach, since for each iteration the optimization variables are set and then the differential equation constraints are integrated. Though straightforward, this approach is generally inefficient because it requires the accurate solution of the model equations at each iteration within the optimization, even when iterates are far from the final optimal solution. 

The simultaneous solution strategy offers several advantages over the sequential approach. A wide range of constraints may be easily incorporated and the solution of the optimization problem provides useful sensitivity information at little additional cost. On the other hand, the sequential approach is straightforward to implement and also has the advantage of well-developed error control. Error control for numerical integrators (used in the sequential approach) is relatively mature when compared, for example, to that of orthogonal collocation on finite elements (a possible technique for a simultaneous approach). 

Optimization running multiple simulations and testing all combinations of strategies to find the optimal solution. 

Given this political history of hydropower in Switzerland, a standard for sustainable hydropower operation was likely to provoke major political reactions. However, expectations associated with market liberalization also partly motivated actors to reconsider their original interest positions. However, solutions could not be found because the interlocked nature of the decision problem represented a classical social dilemma [11]. A social dilemma is present if decisions of two actors depend on each other and if both actors are forced to select a sub-optimal strategy of conduct to minimize their potential losses. An optimal solution would only be realized if each party could tmst the other. 

According to these results the slack variables vanish in the constraints 1 and 2, which are of type <. Therefore, the optimal solution is on the boundary defined by these two constraints. Such constraints are said to be active ones. In physical terms it means that the available supplies of raw material A and are both exhausted. The optimal strategy is producing 16.7 units of product I and 29.3 units of product II. 

Applying the above algorithmic strategy, the optimal solution found, shown in Figure 9.6, has a total annual cost of 156,710. It features (i) task II first and then task I, (ii) a total by-pass of the feed to the two products, and (iii) the bottoms of task II is directed to product P2. 

S. Papageorgaki and G. V. Reklaitis. Optimal design of multipurpose batch plants 2. A decomposition solution strategy. I EC Res., 29(10) 2062,1990. 

By using computer simulation accurate near-optimal solutions can be found with relatively reduced effort quickly by the sequencing of zeotropic mixtures. The following strategy can be applied ... 

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