A large variety of quantitative decision problems in the applied sciences, engineering and economics can be described by constrained optimization models. In these models, the best decision is sought that satisfies all stated feasibility constraints and minimizes (or maximizes) the value of a given objective function.

While man-made objects and (manufacturing, transportation, distribution, etc.) systems often have an approximately linear structure, many other (physical, chemical, biological, geological, environmental, economic, financial, social) systems are typically characterized (also) by nonlinear functional relations. To illustrate this point, one can think of descriptive system models defined by polynomials, exponential and logarithmic functions, trigonometric functions, special functions, integrals, systems of differential equations, stochastic simulation models, or other computational procedures.

As a result, the corresponding nonlinear decision models frequently possess multiple optima of different quality. In such cases, the traditional repertoire of (local) numerical optimization does not guarantee the correct approximation of the globally best solution. This can lead to more costly decisions, inferior design and operations, higher than necessary risk, and so on.

The objective of global optimization is to find the absolutely best solution of nonlinear decision models, in the possible presence of multiple locally optimal solutions.

• min f(x)
  
• g(x) <= 0
  
• a <= x <= b.
  

• x is a real n-vector that describes the possible decisions
  
• a, b are finite, component-wise vector bounds regarding x
  
• f(x) is a continuous function that describes the key model objective
  
• g(x) is a continuous m-vector function that describes the model constraints; the corresponding inequality is interpreted component-wise.

LGO Solver Suite: Key Features

The program system LGO - originally abbreviating a Lipschitz(-continuous) Global Optimizer - assists in the formulation and solution of the broad class of decision problems encompassed by the model form stated above, under 'minimal' analytical assumptions. The general structure postulated makes LGO directly - and easily - applicable to a broad variety of real-life decision problems.

LGO is particularly suitable to analyze design and operational decisions which are related to complete stand-alone ('black box') systems, or to models which are supported by limited, difficult-to-use, confidential, or - due to ongoing development - often changing analytical information.

LGO integrates a suite of robust and efficient global and local scope solvers. These include the following component algorithms:

• adaptive partition and search (branch-and-bound)
  
• adaptive global random search (single-start)
  
• adaptive global random search (multi-start)
  
• constrained local optimization (reduced gradient method).
  

LGO does not require derivative information: the solver operations are based exclusively on the computation of the model function values, at algorithmically selected search points. This feature makes it uniquely suitable to handle models with arbitrary continuous functions, including 'black box' numerical procedures.

Software Implementations

The current nonlinear optimization software implementations - developed with our partners for compiler platforms, spreadsheets, optimization modeling languages, and integrated technical computing systems - are listed below:

• A summary product description in pdf file format
  
  
• LGO solver suite for global and local optimization, with a text I/O interface (for C/C++ and Fortran compiler platforms) available from PCS
  
  
• LGO solver suite with a Microsoft Windows interface (LGO IDE; for Visual Basic, C/C++, Delphi, and Fortran compiler platforms) available from PCS
  
  
• LGO solver engine for AIMMS users available from Paragon Decision Technology
  
  
• LGO solver engine for AMPL users available from PCS
  
  
• LGO solver engine for GAMS users available from the GAMS Development Corporation
  
  
• Global Optimization Toolbox (LGO solver engine for Maple users) available from Maplesoft
  
  
• LGO solver engine for MATLAB users available from TOMLAB Optimization
  
  
• MathOptimizer Professional (LGO solver engine for Mathematica users) available from PCS and Wolfram Research
  
  
• MathOptimizer (native Mathematica solver engine for Mathematica users) available from PCS and Wolfram Research
  
  
• LGO solver engine for MPL users available from Maximal Software