An algorithm for solving large-scale nonlinear ' programs with linear constraints is presented. The method combines efficient sparse-matrix techniques as in the revised simplex method with stable quasi-Newton methods for handling the nonlinearities. A general-purpose production code (MINOS) is described, along with computational experience on a wide variety of problems.

In this paper, the deterministic global optimization algorithm, αBB, (α-based Branch and Bound) is presented. This algorithm offers mathematical guarantees for convergence to a point arbitrarily close to the global minimum for the large class of twice-differentiable NLPs. The key idea is the construction of a converging sequence of upper and lower bounds on the global minimum through the convex relaxation of the original problem. This relaxation is obtained by (i) replacing all nonconvex terms of special structure (i.e., bilinear, trilinear, fractional, fractional trilinear, univariate concave) with customized tight convex lower bounding functions and (ii) by utilizing some α parameters as defined by Maranas and Floudas (1994b) to generate valid convex underestimators for nonconvex terms of generic structure. In most cases, the calculation of appropriate values for the α parameters is a challenging task. A number of approaches are proposed, which rigorously generate a set of α par...

In this paper we describe a new version of a sequential equality constrained quadratic programming method for general nonlinear programs with mixed equality and inequality constraints. Compared with an older version [34] it is much simpler to implement and allows any kind of changes of the working set in every step. Our method relies on a strong regularity condition. As far as it is applicable the new approach is superior to conventional SQP-methods, as demonstrated by extensive numerical tests.

Part I of this paper (Adjiman et al., 1997) described the theoretical foundations of a global optimization algorithm, the ffBB algorithm, which can be used to solve problems belonging to the broad class of twice-differentiable NPLs. For any such problem, the ability to automatically generate progressively tighter convex lower bounding problems at each iteration guarantees the convergence of the branchand -bound ffBB algorithm to within ffl of the global optimum solution. Several methods were presented for the construction of convex valid underestimators for general nonconvex functions. In this second part, the performance of the proposed algorithm and its alternative underestimators is studied through their application to a variety of problems. An implementation of the ffBB is described and a number of rules for branching variable selection and variable bound updates are shown to enhance convergence rates. A user-friendly parser facilitates problem input and provides flexibility in the...

Abstract: Integrated scientific-technical computing (ISTC) environments play an increasing role in advanced systems modeling and optimization. MathOptimizer Professional (MOP) has been recently developed to solve nonlinear optimization problems formulated in the ISTC system Mathematica. We introduce this software package, and review its key functionality and options. MOP is then used to solve illustrative circle packing problems, including both well-frequented models and a new (more difficult) model-class.

Abstract. We present a review of several software products that serve to analyze and solve highly nonlinear—specifically including global—optimization problems across different hardware and software platforms. The implementations discussed are LGO, as a stand-alone, but compiler-dependent modeling and solver environment; its Excel platform implementation; and MathOptimizer, a native solver package for Mathematica users. The discussion advocates the use of hybrid solution strategies, in the context of practical global optimization. Several non-trivial illustrative applications, taken from the literature or received from users of our software products, are also reviewed.