Derivative-free optimization and filter methods to solve nonlinear constrained problems

In real optimization problems, usually the analytical expression of the objective function is not known, nor its derivatives, or they are complex. In these cases it becomes essential to use optimization methods where the calculation of the derivatives, or the verification of their existence, is not necessary: the Direct Search Methods or Derivative-free Methods are one solution. When the problem has constraints, penalty functions are often used. Unfortunately the choice of the penalty parameters is, frequently, very difficult, because most strategies for choosing it are heuristics strategies. As an alternative to penalty function appeared the filter methods. A filter algorithm introduces a function that aggregates the constrained violations and constructs a biobjective problem. In this problem the step is accepted if it either reduces the objective function or the constrained violation. This implies that the filter methods are less parameter dependent than a penalty function. In this work, we present a new direct search method, based on simplex methods, for general constrained optimization that combines the features of the simplex method and filter methods. This method does not compute or approximate any derivatives, penalty constants or Lagrange multipliers. The basic idea of simplex filter algorithm is to construct an initial simplex and use the simplex to drive the search. We illustrate the behavior of our algorithm through some examples. The proposed methods were implemented in Java.

Delineating environmental groundwater vulnerability and protection zones mapping in fractured rock masses

Hard‐rock watersheds commonly exhibit complex geological bedrock and morphological features. Hydromineral resources have relevant economic value for the thermal spas industry. The present study aims to develop a groundwater vulnerability approach in Caldas da Cavaca hydromineral system (Aguiar da Beira, Central Portugal) which has a thermal tradition that dates back to the late 19th century, and contribute to a better understanding of the hydrogeological conceptual site model. In this work different layers were overlaid, generating several thematic maps to arrive at an integrated framework of several key‐sectors in Caldas da Cavaca site. Thus, to accomplish a comprehensive analysis and conceptualization of the site, a multi‐technical approach was used, such as, field and laboratory techniques, where several data was collected, like geotectonics, hydrology and hydrogeology, hydrogeomorphology, hydrogeophysical and hydrogeomechanical zoning aiming the application of the so‐called DISCO method. All these techniques were successfully performed and a groundwater vulnerability to contamination assessment, based on GOD‐S, DRASTIC‐Fm, SINTACS, SI and DISCO indexes methodology, was delineated. Geographical Information Systems (GIS) technology was on the basis to organise and integrate the geodatabases and to produce all the thematic maps. This multi‐technical approach highlights the importance of groundwater vulnerability to contamination mapping as a tool to support hydrogeological conceptualisation, contributing to better decision‐making of water resources management and sustainability.