Eligius M. T. Hendrix

• PhD
• Professor (Full) at University of Malaga

We consider a simplicial branch and bound Global Optimization algorithm, where the search region is a simplex. Apart from using longest edge bisection, a simplicial partition set can be reduced due to monotonicity of the objective function. If there is a direction in which the objective function is monotone over a simplex, depending on whether the...

The use of e-learning systems has a long tradition, where students can study online helped by a system. In this context, the use of recommender systems is relatively new. In our research project, we investigated various ways to create a recommender system. They all aim at facilitating the learning and understanding of a student. We present a common...

We present a novel relaxation for general nonconvex sparse MINLP problems, called overlapping convex hull relaxation (CHR). It is defined by replacing all nonlinear constraint sets by their convex hulls. If the convex hulls are disjunctive, e.g. if the MINLP is block-separable, the CHR is equivalent to the convex hull relaxation obtained by (standa...

In recent years, an interest appeared in integrating various optimization algorithms in machine learning. We study the potential of ensemble learning in classification tasks and how to efficiently decompose the underlying optimization problem. Ensemble learning has become popular for machine learning applications and it is particularly interesting...

This paper studies a retail inventory system for a perishable product, based on a practical setting in Dutch retail. The product has a fixed shelf life of three days upon delivery at the store and product demand has a weekly pattern, which is stationary over the weeks, but varies over the days of the week. Items of varying age occur in stock. Howev...

This study focuses on exhaustive global optimization algorithms over a simplicial feasible set with simplicial partition sets. Bounds on the objective function value and its partial derivative are based on interval automatic differentiation over the interval hull of a simplex. A monotonicity test may be used to decide to either reject a simplicial...

We present new decomposition methods for globally solving complex optimization and machine learning problems based on a generate-refine-and-solve (GRS) approach using inner and outer approximations. The methods are implemented in the open-source frameworks Decogo and Decolearn. Numerical results for complex nonconvex MINLPs are presented. Furthermo...

We study an inventory control problem of a perishable product with a fixed short shelf life in Dutch retail practice. The demand is non-stationary during the week but stationary over the weeks, with mixed LIFO and FIFO withdrawal. The supermarket uses a service level requirement. A difficulty is that the age-distribution of products in stock is not...

Currently minor improvements in image classification tasks are achieved by significantly increasing model complexity. This trend has been ongoing for the last years and high performance models now usually have millions of parameters. Inspired by Ensemble methods, we investigate the potential of Ensemble Learning (EL) to iteratively extend an ensemb...

Several aspects we elaborated in the development of algorithms for MINLP. Mainly the resource constraint vision and the concept of Column Generation are highlighted.

An interesting question for linear programming (LP) algorithms 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 approach is that of basis deficiency-allowing (BDA) simplex variations, which work with a subset of independent columns of the coefficient matrix in...

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...

The computational challenge offered by many traditional network flow models is modest, and large-scale instances can be solved fast. When the composition of the flow is part of the model, the required computation time may increase substantially. This is in particular true for the pooling problem, where the relative content of certain flow component...

Operators of underground water supply networks are challenged with pipe replacement decisions, because pipes are subject to increased failure rates as they age and financial resources are often limited.We study the optimal replacement time and optimal number of pipe replacements such that the expected failure cost and replacement cost are minimized...

Production planning and scheduling in food processing industry (FPI) requires taking specific characteristics into account. First of all, setups are usually sequence-dependent and may include the so-called non-triangular setup conditions. Secondly, planning problems in FPI must take product decay into consideration. We present an MILP model that ha...

A natural way to define branching in branch and bound (B&B) for blending problems is bisection. The consequence of using bisection is that partition sets are in general irregular. The question is how to use regular simplices in the refinement of the unit simplex. A regular simplex with fixed orientation can be represented by its center and size, fa...

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...

The iterative bisection of the longest edge of the unit simplex generates a binary tree, where the specific shape depends on the chosen longest edges to be bisected. In global optimization, the use of various distance norms may be advantageous for bounding purposes. The question dealt with in this paper is how the size of a binary tree generated by...

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.

A multitude of heuristic stochastic optimization algorithms have been described in literature to obtain good solutions of the box-constrained global optimization problem often with a limit on the number of used function evaluations. In the larger question of which algorithms behave well on which type of instances, our focus is here on the benchmark...

We study the practical decision problem of fresh food production with a long production lead time to decide every period (e.g. week) how many items to produce. When a batch is ready for use, its items have a fixed shelf life, after which the items become waste in the sense that they cannot be sold anymore. The demand for (fresh) food products is un...

En el ámbito de la optimización global basada en técnicas de ramificación y acotación, cuando el espacio de búsqueda es un n-símplex regular es habitual utilizar como regla de división la bisección por el lado mayor, debido a que garantiza la convergencia del algoritmo. Cuando la dimensión del n-símplex es mayor de 2 existen varios lados mayores qu...

Multitud de problemas de ciencia e ingeniería se pueden clasificar como problemas de Optimización Global, donde se pretende seleccionar la mejor solución con respecto a algún criterio, de un conjunto de posibles soluciones. A la hora de implementar un algoritmo para resolver este tipo de problemas, existen dos opciones: empezar desde cero o reutili...

En este trabajo se analizan y evalúan dos implementaciones de un algoritmo de optimización para un problema de control de inventarios de productos perecederos. Las implementaciones se han llevado a cabo utilizando una arquitectura heterogénea donde cada nodo está compuesto por varios multicores y varias GPUs. Las versiones paralelas que se han desa...

The computational challenge offered by most traditional network flow models is modest, and large scale instances can be solved fast. The challenge becomes more serious if the composition of the flow has to be taken into account. This is in particular true for the pooling problem, where the relative content of certain flow components is restricted....