Interior Point Constraints

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)]. 

LP software includes two related but fundamentally different kinds of programs. The first is solver software, which takes data specifying an LP or MILP as input, solves it, and returns the results. Solver software may contain one or more algorithms (simplex and interior point LP solvers and branch-and-bound methods for MILPs, which call an LP solver many times). Some LP solvers also include facilities for solving some types of nonlinear problems, usually quadratic programming problems (quadratic objective function, linear constraints see Section 8.3), or separable nonlinear problems, in which the objective or some constraint functions are a sum of nonlinear functions, each of a single variable, such as... 

Mulligan, A. E., and Ahlfeld, D. P. (1999b). An interior point boundary projection method for nonlinear groundwater optimization with zero-derivative constraints. RCGRD Publication 98-2, University ofVermont, Burlington, VT. 

For single separation duty Farhat et al. (1990) presented multiple criteria decisionmaking (MCDM) NLP based problem formulations for multiperiod optimisation. This involves either maximisation (Problem 1) of specified products (main-cuts) or minimisation (Problem 2) of unspecified products (off-cuts) subject to interior point constraints. These two optimisation problems are described below. 

It is quite obvious that a two level optimisation formulation can be very expensive in terms of computation time. This is due to the fact that for any particular choice of R1 and xRi a complete solution (sub-optimal) of the two distillation tasks are required. The same is true for each gradient evaluation with respect to the decision variables (B7and xRj). Mujtaba (1989) proposed a faster one level dynamic optimisation formulation for the recycle problem which eliminates the requirement to calculate any sub-optimal or intermediate solution. In this formulation the total distillation time is minimised directly satisfying the separation requirements for the first distillation task as interior point constraints and for the second distillation task as final time constraints. It was found that the proposed formulation was much more robust and at least 5 times faster than the classical two level formulation. 

We have used sensitivity equation methods (Leis and Kramer, 1985) for gradient evaluation as these are simple and efficient for problems with few parameters and constraints. In general, the balance in efficiency between sensitivity and adjoint methods depends on the type of problem being addressed. Adjoint methods are particularly advantageous for optimal control problems in which the inputs are represented as a large number of piecewise constant input values and few interior point constraints exist. Sensitivity methods are preferable for problems with few parameters and many constraints. 

The outer-approximation algorithm (Section II.A) took six iterations to identify this solution, with a projection factor, e, of. 05 on the disturbance amplitude. Both vertex and nonvertex constraint maximizers were identified, confirming the need to consider nonvertex maximizers. The variables that contributed nonvertex maximizers were the step switching times (several times) and the measurement lags (once). Robustness was verified with respect to all vertex combinations of uncertain values and a random selection of interior points (ivert = 1, nrand y = 1000). 

Maximization of throughput and minimization of desorbent consumption. e constraint method A superstructure optimization problem for SMB process is considered. An interior point optimizer (IPOPT) is used to solve the single objective subproblems. Kawajiri and Biegler (2006)... 

The definition of the class of very large problems for which interior point methods dominate is changing almost daily as computer implementations of the interior point method become more and more efficient. However, since 1984 there have been dramatic improvements in the computer implementations of the simplex method as well, largely spurred by the competition between the two methods. As a result, there is not currently much reason to prefer either method for LP models with a few hundred constraints. Beyond a few thousand constraints, however, the interior point method leaves the simplex method further and further behind as problem size grows. 

Barrier function methods are very similar to penalty function methods except that they start at an interior point of the feasible region and set a barrier against leaving the feasible region. In this case, the feasible region must have an interior, so this method is generally restricted to inequality constraints. Consider the nonlinear problem with inequality constraints. 

Consider the problem described in Section 6.5 (p. 172), but without the interior point constraints. In this modified problem, if the controls happen to be discontinuous (i.e., have finite jump discontinuities) at time ti, then the states defined by... 

Therefore, the Attic method is a halfway route between the Simplex method and the Interior Point method in fact, the working point must not lay on a vertex, which makes the method behave more like the Interior Point method, but, at the same time, the active constraints are satisfied, which makes it look like the Simplex method. 

Contrary to Simplex and Interior Point, nonnegativity constraints do not play a special role in the Attic method, apart from their prerogative of making the corresponding row of the matrix D particularly sparse. 

In the majority of the literature tests for linear programming, bound constraints are rarely provided and often only nonnegativity bounds are given (this is required by the Simplex and Interior Points methods). 

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