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

Applications

Since 1990, our software products have been used to solve global optimization problems, originating from a broad range of application areas. Currently, models with up to a few thousand decision variables and constraints can be handled on personal computers. The corresponding program execution times could vary significantly, of course (since GO problem instances can be more or less difficult, and model function evaluations can take more or less time).

Our software users have been applying the listed software products, for instance, in the following areas:

• advanced engineering design (acoustics, automotive, electronics, lasers, optics, robotics, surveillance, and other areas)
  
• aggregation of expert opinions (site or facility location, risk assessment)
  
• calibration and operation of laboratory or medical instruments
  
• data classification (cluster analysis)
  
• data visualization, multidimensional scaling and state space reduction
  
• experimental design
  
• extremal energy (potential function) configurations in physical, chemical, and biological modeling
  
• facility location and service allocation (distribution) problems
  
• financial modeling and optimization
  
• general nonlinear approximation
  
• industrial design
  
• marketing research
  
• model fitting to empirical data: identification, calibration and verification procedures
  
• object packing and configuration design in scientific studies and in industrial engineering
  
• optimized design and operation of "black box" (confidential or other completely closed) systems
  
• process optimization in the chemical and energy industries
  
• risk analysis and management
  
• robust product or mixture design in the chemical and processing industries
  
• stochastic simulation and optimization studies in engineering applications
  
• systems of nonlinear equations and inequalities
  
• supply chain management.
  

For additional information on these applications, please see our books and list of publications. Some of the professional links offer further information on global optimization applications.

Thank you for your interest.