# Eligius M. T. HendrixUniversity of Malaga | UMA · Department of Computer Architecture

Eligius M. T. Hendrix

PhD

## About

264

Publications

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2,347

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Introduction

With our co-workers we are working on several interesting algorithms. Questions on monotonicity in simplicial branch and bound algorithms. On deep learning in hyperspectral data classification, but also the potential of using decomposition in learning. Also in MINLP, our question is how to use decomposition to develop efficient algorithms. We started to investigate again Stochastic GO algorithms, specifically their local search behaviour.

Additional affiliations

September 2018 - present

January 2017 - September 2018

June 2015 - December 2016

Education

September 1984 - November 1987

September 1981 - August 1984

## Publications

Publications (264)

We present inner approximations methods for nonconvex MINLP and Ensemble Learning

We consider a simplicial branch and bound (BnB) Global Optimization (GO) algorithm, where the search region is a simplex. Apart from using longest edge bisection, a simplicial partition set can be re-moved/reduced due to monotonicity of the objective function. Here we use Automatic Differentiation over the interval hull of a simplex to get bounds o...

Stochastic Global Optimization has left us with the challenge to derive theoretical results on the speed of convergence. In this contribution , we formulate this challenge towards the question to look at the algorithm from a Markovian perspective. Several considerations are mentioned for the case of Controlled Random Search (CRS).

Optimization and learning problems are becoming increasingly complex and often cannot be solved with traditional deterministic methods , like Branch-and-Bound (BB). We present inner approximation methods for globally solving complex optimization and learning problems based on generating approximation models (master problems) by solving easier sub-p...

Over the last decades, algorithms have been developed for checking copositivity of a matrix. Methods are based on several principles, such as spatial branch and bound, transformation to Mixed Integer Programming, implicit enumeration of KKT points or face-based search. Our research question focuses on exploiting the mathematical properties of the r...

The concept of exploiting proven monotonicity for dimension reduction and elimination of partition sets is well known in the field of Interval Arithmetic Branch and Bound (B &B). Part of the concepts can be applied in simplicial B &B over a box. The focus of our research is here on minimizing a function over a lower simplicial dimension feasible se...

An interesting phenomenon in linear programming (LP) is how to deal with solutions in which the number of nonzero variables is less than the number of rows of the matrix in standard form. An interesting approach is that of basis-deficiency-allowing (BDA) simplex variations, which work with a subset of independent columns of the coefficient matrix i...

Dynamic programming (DP) and specifically Markov Decision Problems (MDP) are often seen in inventory control as a theoretical path towards optimal policies, which are (often) not tractable due to the curse of dimensionality. A careful bounding of decision and state space and use of resources may provide the optimal policy for realistic instances de...

The careful design of experiments in spatial statistics aims at estimating models in an accurate way. In the field of spatial deep learning to classify spatial observations, the training set used to calibrate a model or network is usually determined in a random way in order to obtain a representative sample. This chapter will sketch with examples t...

Energy system optimization models are typically large models which combine sub-models which range from linear to very nonlinear. Column generation (CG) is a classical tool to generate feasible solutions of sub-models, defining columns of
global master problems, which are used to steer the search for a global solution. In this paper, we present a n...

El año 2020 nos sorprendió con la llegada de una pandemia, obligando a los docentes de todas las universidades e instituciones educativas a restrin-gir total o parcialmente las actividades presenciales. Esto trajo consigo la necesidad de cambiar de manera abrupta e inmediata nuestra forma de impartir cla-ses. Había que pasar de un sistema mayoritar...

Use of derivative bounds in simplical branch and bound based on interval arithmetic bounding. The presentation can be found in https://player.vimeo.com/video/601710760?h=9b3aecdc86

In engineering optimization with continuous variables, the use of Stochastic Global Optimization (SGO) algorithms is popular due to the easy availability of codes. All algorithms have a global and local search character, where the global behaviour tries to avoid getting trapped in local optima and the local behaviour intends to reach the lowest obj...

Land-cover information is of paramount importance in a wide range of environmental and socioeconomic applications. Deep learning (DL) provides a large variety of potential models for extracting useful information from raw images. However, remote sensing image (RSI) classification remains a challenging goal due to the intrinsic features of the data,...

Most industrial optimization problems are sparse and can be formulated as block-separable mixed-integer nonlinear programming (MINLP) problems, defined by linking low-dimensional sub-problems by (linear) coupling constraints. This paper investigates the potential of using decomposition and a novel multiobjective-based column and cut generation appr...

Branch and Bound (B&B) algorithms in Global Optimization are used to perform an exhaustive search over the feasible area. One choice is to use simplicial partition sets. Obtaining sharp and cheap bounds of the objective function over a simplex is very important in the construction of efficient Global Optimization B&B algorithms. Although enclosing...

In this talk we present Decogo, a generic software framework for solving sparse nonconvex MINLPs, based on decomposition based successive approximation. Similar as Column Generation (CG) algorithms for solving huge crew scheduling problems, Decogo computes a solution candidate of a MINLP by first computing a solution of a convex hull relaxation (CR...

Simplicial based Global Optimization branch and bound methods require tight bounds on the objective function value. Recently, a renewed interest appears on bound calculation based on Interval Arithmetic by Karhbet and Kearfott (2017) and on exploiting second derivative bounds by Mohand (2021). The investigated question here is how partial derivativ...

Convolutional neural networks (CNNs) are a noteworthy tool for the classification of hyperspectral images (HSIs). CNNs apply non-linear activation functions to learn data patterns.One of them is the rectified linear unit (ReLU), which is apiece-wise linear function with a value which is the input ifpositive and zero otherwise. As a result, it is co...

Inhomogeneous membrane cascade systems have been utilized to purify fructooligosaccharides (FOS). Such a process allows a different setup at every stage of the cascade. Varying the setup at every stage implies an optimization problem related to the selection of the membrane and combinations of operating conditions. This paper solves the optimizatio...

Many industrial optimization problems are sparse and can be formulated as block-separable mixed-integer nonlinear programming (MINLP) problems, where low-dimensional sub-problems are linked by a (linear) knapsack-like coupling constraint. This paper investigates exploiting this structure using decomposition and a resource constraint formulation of...

Detection of surface material based on hyperspectral imaging (HSI) analysis is an important and challenging task in remote sensing. It is widely known that spectral-spatial data exploitation performs better than traditional spectral pixel-wise procedures. Nowadays, convolutional neural networks (CNNs) have shown to be a powerful deep learning (DL)...

Retailers can exploit the consumer willingness to substitute to improve their profit, service level and waste. This paper investigates to what extent such improvement can be realised by the replenishment decisions. Two order policies are compared: one policy neglecting product substitution, and a new policy that decides on order quantities for all...

This paper presents a new two-phase method for solving convex mixed-integer nonlinear programming (MINLP) problems, called Decomposition-based Outer Approximation Algorithm (DECOA). In the first phase, a sequence of linear integer relaxed sub-problems (LP phase) is solved in order to rapidly generate a good linear relaxation of the original MINLP p...

Most industrial optimization problems are sparse and can be formulated as block-separable mixed-integer nonlinear programming (MINLP) problems, defined by linking low-dimensional sub-problems by (linear) coupling constraints. Decomposition methods solve a block-separable MINLP by alternately solving master problems and sub-problems. In practice, de...

Encontrar el mínimo de un problema de optimización cuadrática estándar permite determinar que una matriz simétrica es copositiva. El espacio de búsqueda del valor mínimo se realiza en un simplex unidad. En las jornadas SARTECO'18 se presentó un algoritmo que evalúa las caras de un simplex unidad para determinar si una matriz simétrica es copositi-v...

The efficient control of electrical vehicles may contribute to sustainable use of energy. In recent studies, a model has been analyzed and several algorithms based on branch and bound have been presented. In this work, we discuss a reformulated model on the control of an electric vehicle based on the minimization of the energy consumption during an...

Maintenance costs account for a large part of the total cost of an offshore wind farm. Several models have been presented in the literature to optimize the fleet composition of the required vessels to support maintenance tasks. We provide a mixed integer linear programming (MILP) description of such a model, where on the higher level, the fleet com...

Dynamic programming (DP) approaches, in particular value iteration, is often seen as a method to derive optimal policies in inventory management. The challenge in this approach is to deal with an increasing state space when handling realistic problems. As a large part of world food production is thrown out due to its perishable character, a motivat...

As a large part of world food production spoils/expires before consumption, reduction of food waste by optimizing order policies in retail is of importance. We sketch here the computational burden of trying to obtain the optimal order quantities with the process of Value Iteration for a retailer situation with highly perishable products. It appears...

Branch and bound methods in Global Optimization guarantee to find the set of global minimum points up to a certain accuracy. Partition sets typically take the shape of boxes, cones and simplices. In the tradition of interval arithmetic for generating bounds on box shaped partition sets, the concept of monotonicity test is used in order to reduce di...

Over the last decades checking copositivity of matrices by simplicial subdivision of the unit simplex has made a big progress. Recently it has been shown that surprisingly the use of regular simplicial subdivisions may have some advantage over traditional iterative bisection of simplices. In this contribution, we pose the question whether regular s...

In literature, one can find a branch and bound approach for the control of electric vehicles was published. Using that model, we create a DP implementation to obtain similar results.

Dynamic programming (DP) is often seen in inventory control to lead to optimal ordering policies. When considering stationary demand, Value Iteration (VI) may be used to derive the best policy. In this paper, our focus is on the computational procedures to implement VI. Practical implementation requires bounding carefully the state space and demand...

Traditional deterministic global optimization methods are often based on a Branch-and-Bound (BB) search tree, which may grow rapidly, preventing the method to find a good solution. Motivated by decomposition-based inner approximation (column generation) methods for solving transport scheduling problems with over 100 million variables, we present a...

In retail, it is usual to measure the performance of inventory control by a so-called service level which measures the number of days (probability) that demand is fulfilled for a certain product, i.e. the product is not out of stock. Part of the customers that do not encounter the product in stock will look for a substitute. To measure the performa...

Decision-making often refers to ranking alternatives based on many involved criteria. Since the introduction of the Analytic Hierarchy Process (AHP) in 1980, pairwise comparisons of criteria have a long tradition in multi-criteria decision-making. One of the main concerns of the AHP refers to the inconsistency of decision makers in pairwise compari...

El problema de encontrar el minimo de un problema de optimizacíon cuadrática estándar permite determinar cuando la matriz simmetrica involucrada en la formulaciíon es copositiva. El espacio de búsqueda del valor minimo se realiza en un simplex
unidad. Recientemente se ha desarrollado un algoritmo que evalúa las facetas de un simplex unidad para det...

Una gran parte de la producción mundial de alimentos se desecha debido a su carácter pere-cedero. Esto hace que exista una gran motivación para investigar las políticas óptimas de pedidos de productos perecederos en el comercio minorista que minimicen las grandes cantidades de alimentos que van a la basura. En este trabajo analizamos los aspectos c...

This paper summarizes our findings with respect to order policies for an inventory control problem for a perishable product with a maximum fixed shelf life in a periodic review system, where chance constraints play a role. A Stochastic Programming (SP) problem is presented which models a practical production planning problem over a finite horizon....

Refinement of the unit simplex by iterative longest edge bisection (LEB) up to sub-simplices have a size smaller or equal to a given accuracy, generates a binary tree. For a dimension higher than three, the size of the generated tree depends on the bisected LE. There may exist more than one selection sequence of LE that solves the Smallest Binary T...

This paper studies the dynamic application of the minimax regret (MR) decision criterion to identify robust flood risk management strategies under climate change uncertainty and emerging information. An MR method is developed that uses multiple learning scenarios, for example about sea level rise or river peak flow development, to analyse effects o...

Branch and Bound (B&B) algorithms are known to exhibit an irregularity of the search tree. Therefore, developing a parallel approach for this kind of algorithms is a challenge. The efficiency of a B&B algorithm depends on the chosen Branching, Bounding, Selection, Rejection, and Termination rules. The question we investigate is how the chosen platf...

Maintenance provides a large part of the cost of an offshore wind farm. Several models have been presented in literature to optimize the fleet composition of the required vessels. A drawback such models is that they are based on perfect information on weather and incidences to schedule for the coming year. Our research question is what will happen...

En este artículo exponemos cómo hemos aprovechado el interés que los alumnos demuestran por la plataforma Raspberry Pi para facilitar el estudio de conceptos y técnicas impartidas en varias asignaturas de Ingeniería. Además, proponemos usar esta misma plataforma en distintos cursos de forma que se mejore la coordinación vertical en asignaturas de p...

En Optimización Global, mediante Ramificación y Acotación, es habitual usar la bisección por lado mayor como método de refinamiento cuando el espacio de búsqueda es un símplice. En problemas donde la dimensión es mayor que 3 pueden existir diversos lados mayores y la elección de uno u otro afecta al tamaño del arbol binario que se genera en el refi...

The offshore wind energy industry is expected to continue its growth tendency in the near future. The European Wind Energy Association expects in its Central Scenario by 2030 a total installed capacity of 66 GW of offshore wind in the UE. Offshore wind farms (OWFs) are large scale infrastructures, requiring maintenance fleets to perform operations...

Branch and bound (BnB) Global Optimization algorithms can be used to find the global optimum (minimum) of a multiextremal function over the unit hypercube and unit simplex with a guaranteed accuracy. Subdivision strategies can take the information of the evaluated points into account leading to irregular shaped subsets. This study focuses on the pa...

This article describes the process of making use of the interest of students for the Raspberry Pi platform to facilitate study of concepts and techniques in several Engineering courses. Moreover, we show how this platform was used to improve the vertical coordination in the Computer Engineering study from 1 st to 4 th year in the University of Mála...

We present a discrete optimisation model that chooses an optimal fleet of vessels to support maintenance operations at Offshore Wind Farms (OFWs). The model is presented as a bi-level problem. On the first (tactical) level, decisions are made on the fleet composition for a certain time horizon. On the second (operational) level, the fleet is used t...

In this paper, we study the single-item single-stocking location non-stationary stochastic lot sizing problem for a perishable product. We consider fixed and proportional ordering cost, holding cost and penalty cost. The item features a limited shelf life, therefore we also take into account a variable cost of disposal. We derive exact analytical e...

Management objectives of the European Union for North Sea fish stocks are shifting towards considering both biological sustainability and economic benefits. As part of multiannual management plans, an adjustment restriction on fish quota has been introduced. Its objective is to obtain an efficient fish stock and to reduce overcapacity for the fishi...

This paper analyses and evaluates parallel implementations of an optimization algorithm for perishable inventory control problems. This iterative algorithm has high computational requirements when solving large problems. Therefore, the use of parallel and distributed computing reduces the execution time and improves the quality of the solutions. Th...

Traffic lights are put in place to dynamically change priority between traffic participants. Commonly, the duration of green intervals and the grouping, and ordering in which traffic flows are served are pre-fixed. In this chapter, the problem of minimizing vehicle delay at isolated intersections is formulated as a Markov Decision Process (MDP). So...

This paper describes and analyses a bi-level Markov Decision Problem (MDP). The model has been used to study questions on the setting of fisheries quota. The problem extends earlier models in literature and describes fish stock and economic dynamics. At the first level, an authority decides on the quota to be fished keeping in mind long term revenu...

In several areas like global optimization using branch-and-bound methods for mixture design, the unit n-simplex is refined by longest edge bisection (LEB). This process provides a binary search tree. For (Formula presented.), simplices appearing during the refinement process can have more than one longest edge (LE). The size of the resulting binary...

Wind energy can lead to a great socioeconomic impact in Europe nowadays. Therefore, it has gained more attention lately. Eolic energy is generated by a so-called wind farm, i.e. a set of wind turbines. The wind turbines have to be connected via cables into a power grid and the generated power is collected at the main power station. In this investig...

In several areas like Global Optimization using branch-and-bound (B&B) methods for mixture design , the unit n-simplex is refined by longest edge bisection. This process provides a binary search tree. For n > 2, simplices appearing during the refinement process can have more than one Longest Edge (LE). The size of the resulting binary tree depends...

Climate control is essential for managing the production of horticulture in greenhouses. One of the traditional ways to control the climate in the greenhouse is Model Predictive Control (MPC). This paper studies the potential of using branch and bound algorithms in order to effectively reach the best control in MPC.

Over the last decades checking copositivity of matrices by simplicial subdivision of the unit simplex has made a big progress. Recently it has been showing that surprisingly the use of regular sim-plicial subdivisions may have some advantage over traditional iterative bisection of simplices. In this contribution we pose the question whether regular...

The blending problem is studied as a problem of finding cheap robust feasible solutions on the unit simplex. Usually longest edge bisection is used in the simplex refinement process, because it is an easy division method that guarantees finding a solution. However, with this method more than one longest edge could appear in the simplex and the beha...

In Branch and Bound (BnB) algorithms, the branching rule plays an important rule in order to reduce the number of evaluated sub-problems and points. Recent studies addressed the unit simplex refinement with regular simplices. When the achieved accuracy is related on the distance among sampled points, a common method is to stop the refinement when t...