Local optimization algorithms

We emphasize that in addition to a global optimization strategy, the precise determination of the structure requires an efficient and reliable local optimization algorithm. MD does not provide such a tool because it cannot optimize efficiently the degrees of freedom... 

Step 3 Solve the convex lower bounding problem using a local optimization algorithm (e.g. MINOS [21], NPSOL [22]) which provides a lower bound for the solution of the original problem. 

The basic structure of an iterative local optimization algorithm is one of greedy descent . It is based on one of the following two algorithmic frameworks line-search or trust-region methods. Both are found throughout the literature and in software packages and are essential components of effective... 

The line search and the procedure that defines p t form the central components of the basic descent local optimization algorithm above. The work in the line search (number of polynomial interpolations) should be balanced with the overall progress realized in the minimization algorithm. 

We start with continuous variable optimization and consider in the next section the solution of NLP problems with differentiable objective and constraint functions. If only local solutions are required for the NLP problem, then very efficient large-scale methods can be considered. This is followed by methods that are not based on local optimality criteria we consider direct search optimization methods that do not require derivatives as well as deterministic global optimization methods. Following this, we consider the solution of mixed integer problems and outline the main characteristics of algorithms for their solution. Finally, we conclude with a discussion of optimization modeling software and its implementation on engineering models. 

This basic concept leads to a wide variety of global algorithms, with the following features that can exploit different problem classes. Bounding strategies relate to the calculation of upper and lower bounds. For the former, any feasible point or, preferably, a locally optimal point in the subregion can be used. For the lower bound, convex relaxations of the objective and constraint functions are derived. 

Dealing with Z BZ directly has several advantages if n — m is small. Here the matrix is dense and the sufficient conditions for local optimality require that Z BZ be positive definite. Hence, the quasi-Newton update formula can be applied directly to this matrix. Several variations of this basic algorithm... 

Other decision rules are, of course, possible. As noted earlier, local minima in the objective function surface tend to trap optimization algorithms into globally nonoptimum solutions. When an algorithm is trapped, small perturbations in the estimate elements o(xn) only raise the value of the objective function. .., d(xM)]. In conventional optimization algo-... 

Remark 1 The mathematical model is an MINLP problem since it has both continuous and binary variables and nonlinear objective function and constraints. The binary variables participate linearly in the objective and logical constraints. Constraints (i), (iv), (vii), and (viii) are linear while the remaining constraints are nonlinear. The nonlinearities in (ii), (iii), and (vi) are of the bilinear type and so are the nonlinearities in (v) due to having first-order reactions. The objective function also features bilinear and trilinear terms. As a result of these nonlinearities, the model is nonconvex and hence its solution will be regarded as a local optimum unless a global optimization algorithm is utilized. 

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