Software Development

### Global Optimization

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 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.

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.

### The Global Optimization Model

The general global optimization model can be concisely formulated as shown below.

• min f(x)
• g(x) <= 0
• a <= x <= b.
In the model formulation the following notatons are used:
• 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 defines the model objective
• g(x) is a continuous m-vector function that defines the model constraints; the corresponding vector inequality is interpreted component-wise.

### LGO Solver Suite: Key Features

LGO - abbreviating a Lipschitz-continuous Global Optimizer, named after one of its key solver components - serves to solve instances from 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 related to complete stand-alone 'black box' systems, and to models which are based on 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 optimization strategies. These include the following component algorithms:

• a combination of heuristic global presolvers
• partition and search (continuous branch-and-bound)
• global random search (single-start)
• 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 LGO uniquely suitable to handle models with arbitrary continuous functions, including 'black box' numerical procedures.

### Software Implementations

Please see the main page. You can also download a summary description of the currently available software product versions.

### Applications

Our software products have been used by our worldwide clientele to solve global (and local) optimization problems which originate from a broad range of application areas. Currently, models with up to several 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 clients have been applying LGO and the related range of our software products, for instance, in the following areas:

• advanced engineering design (acoustics, automotive, electronics, lasers, optics, robotics, space, 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 analysis and visualization
• experimental design
• 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
• modeling and optimization studies in defence
• 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