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Evald Ubi

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Tallinn University of TechnologyAssociate professor

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Steven J. Miller

Williams College

Arizona State University

Samit Chakrabarty

University of Leeds

Viacheslav Kuleshov

Stockholm University

Udochukwu Akuru

University of Nigeria, Nsukka

Adebayo D Agunbiade

Olabisi Onabanjo University, Ago-Iwoye, Nigeria

Kumar Biswajit Debnath

Heriot-Watt University

National Taiwan University

National Physical Laboratory

Prof.dr. Erkan Isigicok

Uludag University

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Papers by Evald Ubi

Mathematical programming via the least-squares method

Central European Journal of Mathematics, 2010

The least-squares method is used to obtain a stable algorithm for a system of linear inequalities... more The least-squares method is used to obtain a stable algorithm for a system of linear inequalities as well as linear and nonlinear programming. For these problems the solution with minimal norm for a system of linear inequalities is found by solving the non-negative least-squares (NNLS) problem. Approximate and exact solutions of these problems are discussed. Attention is mainly paid to finding the initial solution to an LP problem. For this purpose an NNLS problem is formulated, enabling finding the initial solution to the primal or dual problem, which may turn out to be optimal. The presented methods are primarily suitable for ill-conditioned and degenerate problems, as well as for LP problems for which the initial solution is not known. The algorithms are illustrated using some test problems.

A version of the simplex method for solving linear systems of inequalities and linear programming problems

arXiv (Cornell University), Apr 24, 2023

In order to find a non-negative solution to a system of inequalities, the corresponding dual prob... more In order to find a non-negative solution to a system of inequalities, the corresponding dual problem is composed, which has a suitable unity basic matrix. In such a formulation, the objective function is replaced by set of constraints based on that function. Additional constraints can be used for accelerating calculations in the first phase of the simplex method. As an example the solution of the Klee-Minty hypercube problem is described in detail.

СТАТИСТИЧЕСКОЕ ИССЛЕДОВАНИЕ И МЕТОД РЕШЕНИЯ ЗАДАЧ СТОХАСТИЧЕСКОГО ПРОГРАММИРОВАНИЯ

Proceedings of the Academy of Sciences of the Estonian SSR. Physics. Mathematics, 1977

ИСПОЛЬЗОВАНИЕ МЕТОДА НАИМЕНЬШИХ КВАДРАТОВ В МАТЕМАТИЧЕСКОМ ПРОГРАММИРОВАНИИ

Proceedings of the Academy of Sciences of the Estonian SSR. Physics. Mathematics, 1989

On stable least squares solution to the system of linear inequalities

Central European Journal of Mathematics, 2007

The system of inequalities is transformed to the least squares problem on the positive ortant. Th... more The system of inequalities is transformed to the least squares problem on the positive ortant. This problem is solved using orthogonal transformations which are memorized as products. Author's previous paper presented a method where at each step all the coefficients of the system were transformed. This paper describes a method applicable also to large matrices. Like in revised simplex method, in this method an auxiliary matrix is used for the computations. The algorithm is suitable for unstable and degenerate problems primarily.

Exact and stable least squares solution to the linear programming problem

Central European Journal of Mathematics, 2005

A linear programming problem is transformed to the finding an element of polyhedron with the mini... more A linear programming problem is transformed to the finding an element of polyhedron with the minimal norm. According to A.Cline [6], the problem is equivalent to the least squares problem on positive ortant. An orthogonal method for solving the problem is used. This method was presented earlier by the author and it is based on the highly developed least squares technique. First of all, the method is meant for solving unstable and degenerate problems. A new version of the artifical basis method (M-method) is presented. Also, the solving of linear inequality systems is considered.

НАХОЖДЕНИЕ ОПТИМАЛЬНОГО РАНДОМИЗИРОВАННОГО РЕШЕНИЯ В M- И P-МОДЕЛЯХ СТОХАСТИЧЕСКОГО ПРОГРАММИРОВАНИЯ

МЕТОД ОРТОГОНАЛЬНЫХ ПРЕОБРАЗОВАНИЙ

Proceedings of the Academy of Sciences of the Estonian SSR. Physics. Mathematics, 1989

An analysis of community structure detection for educational coopetition

2013 Second International Conference on E-Learning and E-Technologies in Education (ICEEE), 2013

ABSTRACT The goal of the paper is to study how the strictly optimal solutions of community detect... more ABSTRACT The goal of the paper is to study how the strictly optimal solutions of community detection, based on similarity matrices, depend on the parameter of the distance threshold setting method, applied beforehand. In order to detect communities, we apply the oft-used modularity metric and arrive at strict optimality by linear programming, solving an np-hard problem. The distance threshold method is used, making the matrix more and more sparse, and thus the best value of the threshold is determined, by analyzing the number of subsequent clusters detected. Our method is applied on educational coopetition data in the business school of TUT, with four specializations, out of which we sample 36 students, each time selecting from a pair of specializations. Since the optimal number of clusters tends to be four, for any two-fold sampling, we detect a natural division within each specialization as well, the reason for which is a matter for further study. As a result, coopetition - the simultaneous competition and cooperation - is measured between the departments of the business school. The average grade of the students is a proxy for the competitive score of the department. The traditional conductance is used as a proxy for the cooperative score of the department. For our data, the optimal value for the threshold in community detection is 0.07, this way enough noise has been removed from the data, but not too many values, so that vital information is retained. Thus, we most often obtain our goal of detecting four clusters, in the two-fold sampling, effectively displaying the usefulness of fine-tuning the distance threshold while evaluating it by the strictly optimal community detection.

Predicting Student Retention by Linear Programming Discriminant Analysis

International Journal of Technology and Educational Marketing, 2014

The goal of the paper is to predict student retention with an ensemble method by combining linear... more The goal of the paper is to predict student retention with an ensemble method by combining linear programming (LP) discriminant analysis approaches together with bootstrapping and feature salience detection. In order to perform discriminant analysis, we linearize a fractional programming method by using Charnes-Cooper transformation (CCT) and apply linear programming, while comparing with an approach that uses deviation variables (DV) to tackle a similar multiple criteria optimization problem. We train a discriminatory hyperplane family and make the decision based on the average of the histograms created, thereby reducing variability of predictions. Feature salience detection is performed by using the peeling method, which makes the selection based on the proportion of variance explained in the correlation matrix. While the CCT method is superior in detecting true-positives, DV method excels in finding true-negatives. The authors obtain optimal results by selecting either all 14 (CC...

Data mining the MNC like internal co-opetition duality in a university context

ArXiv, 2012

The goal of the paper is to quantify the simultaneous competition and cooperation that takes plac... more The goal of the paper is to quantify the simultaneous competition and cooperation that takes place in organizations. As the concepts seem to be dichotomous opposites at first, the term internal coopetition duality is put forth. Parallels are drawn between coopetitive processes in big multinational corporations (MNCs) and these taking place in universities, also the structural solutions used in both are analyzed. Data mining is used while looking at how specializations inside the university are in competition for better students. We look at the profiles that students have and find natural divisions between the specializations, by applying graph theory and modularity algorithms for community detection. The competitive position of the specializations is evident in the average grades of the detected communities. The ratio of intercommunity ties to intracommunity ties (conductance) quantifies the cooperative stance, though, as students with similar profiles express linkages in the curric...

A three-phase simplex method for infeasible and unbounded linear programming problems

A three-phase simplex method for infeasible and unbounded linear programming problems, 2020

The paper presents a modified artificial basis method MODART, which combine a big-M method with t... more The paper presents a modified artificial basis method MODART, which combine a big-M method with two-phase method. Unlike previous works, the sum of artificial variables in the objective function has not been used, but an additional constraint has been composed for this sum. In contrast with the classical implementation of the simplex method, in the Phase-0, we take into account the objective function of the initial problem and use the big-M method idea for the enlarged problem. If the solution found is infeasible for the initial problem, then Phase1 finds the feasible solution and Phase-2 the optimal solution. In this work, unbounded, infeasible and degenerate problems are mainly considered. Finally, in part five, suggested formulation and solution to the problems is given together with some computational experience. The main ideas are explained by simple examples. Over 60 years of age problem with M has been finally solved in present paper.

How to choose a pivot element in simplex method

New computa.tiona.l rules ofthe simplex method are represented. They differ from classical rules ... more New computa.tiona.l rules ofthe simplex method are represented. They differ from classical rules in the sense that the column corresponding to the objective function is also transfo.rmed and first the pivot row and then the pivot column is determined. In this case the most nega.tive element in pivot row can be chosen for pivot element. In evaluating procedure systems equivalent to systems appearing in classical simplex method are used and theoretically they determine • the-same sequence of basic solutions. The calcula.tions are more precise due to such choice of pivot element and it is assured also by the results of test-problems. Key woz:da: pivot element, simplex method, ill-conditioned problem of linear programmiJl&. 1. Description of the algorithm of the .implex method. Let us consider the linear programming problem in the following form

A numerically stable least squares solution to the quadratic programming problem

The strictly convex quadratic programming problem is transformed to the least distance problem — ... more The strictly convex quadratic programming problem is transformed to the least distance problem — finding the solution of minimum norm to the system of linear inequalities. This problem is equivalent to the linear least squares problem on the positive orthant. It is solved using orthogonal transformations, which are memorized as products. Like in the revised simplex method, an auxiliary matrix is used for computations. Compared to the modified-simplex type methods, the presented dual algorithm QPLS requires less storage and solves ill-conditioned problems more precisely. The algorithm is illustrated by some difficult problems.

Data mining the MNC like internal co-opetition duality in a university context

The goal of the paper is to quantify the simultaneous competition and cooperation that takes plac... more The goal of the paper is to quantify the simultaneous competition and cooperation that takes place in organizations. As the concepts seem to be dichotomous opposites at first, the term internal coopetition duality is put forth. Parallels are drawn between coopetitive processes in big multinational corporations (MNCs) and these taking place in universities, also the structural solutions used in both are analyzed. Data mining is used while looking at how specializations inside the university are in competition for better students. We look at the profiles that students have and find natural divisions between the specializations, by applying graph theory and modularity algorithms for community detection. The competitive position of the specializations is evident in the average grades of the detected communities. The ratio of intercommunity ties to intracommunity ties (conductance) quantifies the cooperative stance, though, as students with similar profiles express linkages in the curricula; and the choices regarding career development undertaken become evident. Managerial implications discussed include the imperative for actively managing and financially rewarding the outcomes of the coopetitive duality.

Mathematical programming via the least-squares method

Central European Journal of Mathematics, Jul 24, 2010

The least-squares method is used to obtain a stable algorithm for a system of linear inequalities... more The least-squares method is used to obtain a stable algorithm for a system of linear inequalities as well as linear and nonlinear programming. For these problems the solution with minimal norm for a system of linear inequalities is found by solving the non-negative least-squares (NNLS) problem. Approximate and exact solutions of these problems are discussed. Attention is mainly paid to finding the initial solution to an LP problem. For this purpose an NNLS problem is formulated, enabling finding the initial solution to the primal or dual problem, which may turn out to be optimal. The presented methods are primarily suitable for ill-conditioned and degenerate problems, as well as for LP problems for which the initial solution is not known. The algorithms are illustrated using some test problems.

Linear inequalities via least squares

Proceedings of the Estonian Academy of Sciences, 2013

Application of Orthogonal Transformations in the Revised Simplex Method

Unlike the classical method, orthogonal transformations are used for computing more precise simpl... more Unlike the classical method, orthogonal transformations are used for computing more precise simplex tables. These transformations are stored as products. With these transformations the basic matrix is turned to the triangular form, which is used instead of the reciprocal matrix. The values of initial and dual variables are found from the triangular system. The performed algorithm MULT is appropriate for

Predicting student retention by comparing histograms of bootstrapping for Charnes-Cooper transformationlinear programming discriminant analysis

2013 Second International Conference on E-Learning and E-Technologies in Education (ICEEE), 2013

ABSTRACT The goal of the paper is to predict student retention by using linear discriminant analy... more ABSTRACT The goal of the paper is to predict student retention by using linear discriminant analysis with bootstrapping. The result (93%) provides accuracy superior to the bootstrapping of a comparative method, as well as to the non-bootstrapping variations. In order to perform discriminant analysis, we linearize a fractional programming method by using Charnes-Cooper transformation and apply linear programming, while the comparative approach uses deviation variables to tackle a similar multiple criteria optimization problem. We train the discriminatory hyperplane family and apply it to the testing set - thus arriving at a set of histograms. We analyze the histograms by using the simple mean - best for prediction - and a five-fold Kolmogorov-Smirnov test - best used for resources allocation, in order to act on the final results. Final results are the outcome of applying the hyperplane family on freshman data.

Mathematical programming via the least-squares method

Central European Journal of Mathematics, 2010

The least-squares method is used to obtain a stable algorithm for a system of linear inequalities... more The least-squares method is used to obtain a stable algorithm for a system of linear inequalities as well as linear and nonlinear programming. For these problems the solution with minimal norm for a system of linear inequalities is found by solving the non-negative least-squares (NNLS) problem. Approximate and exact solutions of these problems are discussed. Attention is mainly paid to finding the initial solution to an LP problem. For this purpose an NNLS problem is formulated, enabling finding the initial solution to the primal or dual problem, which may turn out to be optimal. The presented methods are primarily suitable for ill-conditioned and degenerate problems, as well as for LP problems for which the initial solution is not known. The algorithms are illustrated using some test problems.

A version of the simplex method for solving linear systems of inequalities and linear programming problems

arXiv (Cornell University), Apr 24, 2023

In order to find a non-negative solution to a system of inequalities, the corresponding dual prob... more In order to find a non-negative solution to a system of inequalities, the corresponding dual problem is composed, which has a suitable unity basic matrix. In such a formulation, the objective function is replaced by set of constraints based on that function. Additional constraints can be used for accelerating calculations in the first phase of the simplex method. As an example the solution of the Klee-Minty hypercube problem is described in detail.

СТАТИСТИЧЕСКОЕ ИССЛЕДОВАНИЕ И МЕТОД РЕШЕНИЯ ЗАДАЧ СТОХАСТИЧЕСКОГО ПРОГРАММИРОВАНИЯ

Proceedings of the Academy of Sciences of the Estonian SSR. Physics. Mathematics, 1977

ИСПОЛЬЗОВАНИЕ МЕТОДА НАИМЕНЬШИХ КВАДРАТОВ В МАТЕМАТИЧЕСКОМ ПРОГРАММИРОВАНИИ

Proceedings of the Academy of Sciences of the Estonian SSR. Physics. Mathematics, 1989

On stable least squares solution to the system of linear inequalities

Central European Journal of Mathematics, 2007

The system of inequalities is transformed to the least squares problem on the positive ortant. Th... more The system of inequalities is transformed to the least squares problem on the positive ortant. This problem is solved using orthogonal transformations which are memorized as products. Author's previous paper presented a method where at each step all the coefficients of the system were transformed. This paper describes a method applicable also to large matrices. Like in revised simplex method, in this method an auxiliary matrix is used for the computations. The algorithm is suitable for unstable and degenerate problems primarily.

Exact and stable least squares solution to the linear programming problem

Central European Journal of Mathematics, 2005

A linear programming problem is transformed to the finding an element of polyhedron with the mini... more A linear programming problem is transformed to the finding an element of polyhedron with the minimal norm. According to A.Cline [6], the problem is equivalent to the least squares problem on positive ortant. An orthogonal method for solving the problem is used. This method was presented earlier by the author and it is based on the highly developed least squares technique. First of all, the method is meant for solving unstable and degenerate problems. A new version of the artifical basis method (M-method) is presented. Also, the solving of linear inequality systems is considered.

НАХОЖДЕНИЕ ОПТИМАЛЬНОГО РАНДОМИЗИРОВАННОГО РЕШЕНИЯ В M- И P-МОДЕЛЯХ СТОХАСТИЧЕСКОГО ПРОГРАММИРОВАНИЯ

МЕТОД ОРТОГОНАЛЬНЫХ ПРЕОБРАЗОВАНИЙ

Proceedings of the Academy of Sciences of the Estonian SSR. Physics. Mathematics, 1989

An analysis of community structure detection for educational coopetition

2013 Second International Conference on E-Learning and E-Technologies in Education (ICEEE), 2013

ABSTRACT The goal of the paper is to study how the strictly optimal solutions of community detect... more ABSTRACT The goal of the paper is to study how the strictly optimal solutions of community detection, based on similarity matrices, depend on the parameter of the distance threshold setting method, applied beforehand. In order to detect communities, we apply the oft-used modularity metric and arrive at strict optimality by linear programming, solving an np-hard problem. The distance threshold method is used, making the matrix more and more sparse, and thus the best value of the threshold is determined, by analyzing the number of subsequent clusters detected. Our method is applied on educational coopetition data in the business school of TUT, with four specializations, out of which we sample 36 students, each time selecting from a pair of specializations. Since the optimal number of clusters tends to be four, for any two-fold sampling, we detect a natural division within each specialization as well, the reason for which is a matter for further study. As a result, coopetition - the simultaneous competition and cooperation - is measured between the departments of the business school. The average grade of the students is a proxy for the competitive score of the department. The traditional conductance is used as a proxy for the cooperative score of the department. For our data, the optimal value for the threshold in community detection is 0.07, this way enough noise has been removed from the data, but not too many values, so that vital information is retained. Thus, we most often obtain our goal of detecting four clusters, in the two-fold sampling, effectively displaying the usefulness of fine-tuning the distance threshold while evaluating it by the strictly optimal community detection.

Predicting Student Retention by Linear Programming Discriminant Analysis

International Journal of Technology and Educational Marketing, 2014

The goal of the paper is to predict student retention with an ensemble method by combining linear... more The goal of the paper is to predict student retention with an ensemble method by combining linear programming (LP) discriminant analysis approaches together with bootstrapping and feature salience detection. In order to perform discriminant analysis, we linearize a fractional programming method by using Charnes-Cooper transformation (CCT) and apply linear programming, while comparing with an approach that uses deviation variables (DV) to tackle a similar multiple criteria optimization problem. We train a discriminatory hyperplane family and make the decision based on the average of the histograms created, thereby reducing variability of predictions. Feature salience detection is performed by using the peeling method, which makes the selection based on the proportion of variance explained in the correlation matrix. While the CCT method is superior in detecting true-positives, DV method excels in finding true-negatives. The authors obtain optimal results by selecting either all 14 (CC...

Data mining the MNC like internal co-opetition duality in a university context

ArXiv, 2012

The goal of the paper is to quantify the simultaneous competition and cooperation that takes plac... more The goal of the paper is to quantify the simultaneous competition and cooperation that takes place in organizations. As the concepts seem to be dichotomous opposites at first, the term internal coopetition duality is put forth. Parallels are drawn between coopetitive processes in big multinational corporations (MNCs) and these taking place in universities, also the structural solutions used in both are analyzed. Data mining is used while looking at how specializations inside the university are in competition for better students. We look at the profiles that students have and find natural divisions between the specializations, by applying graph theory and modularity algorithms for community detection. The competitive position of the specializations is evident in the average grades of the detected communities. The ratio of intercommunity ties to intracommunity ties (conductance) quantifies the cooperative stance, though, as students with similar profiles express linkages in the curric...

A three-phase simplex method for infeasible and unbounded linear programming problems

A three-phase simplex method for infeasible and unbounded linear programming problems, 2020

The paper presents a modified artificial basis method MODART, which combine a big-M method with t... more The paper presents a modified artificial basis method MODART, which combine a big-M method with two-phase method. Unlike previous works, the sum of artificial variables in the objective function has not been used, but an additional constraint has been composed for this sum. In contrast with the classical implementation of the simplex method, in the Phase-0, we take into account the objective function of the initial problem and use the big-M method idea for the enlarged problem. If the solution found is infeasible for the initial problem, then Phase1 finds the feasible solution and Phase-2 the optimal solution. In this work, unbounded, infeasible and degenerate problems are mainly considered. Finally, in part five, suggested formulation and solution to the problems is given together with some computational experience. The main ideas are explained by simple examples. Over 60 years of age problem with M has been finally solved in present paper.

How to choose a pivot element in simplex method

New computa.tiona.l rules ofthe simplex method are represented. They differ from classical rules ... more New computa.tiona.l rules ofthe simplex method are represented. They differ from classical rules in the sense that the column corresponding to the objective function is also transfo.rmed and first the pivot row and then the pivot column is determined. In this case the most nega.tive element in pivot row can be chosen for pivot element. In evaluating procedure systems equivalent to systems appearing in classical simplex method are used and theoretically they determine • the-same sequence of basic solutions. The calcula.tions are more precise due to such choice of pivot element and it is assured also by the results of test-problems. Key woz:da: pivot element, simplex method, ill-conditioned problem of linear programmiJl&. 1. Description of the algorithm of the .implex method. Let us consider the linear programming problem in the following form

A numerically stable least squares solution to the quadratic programming problem

The strictly convex quadratic programming problem is transformed to the least distance problem — ... more The strictly convex quadratic programming problem is transformed to the least distance problem — finding the solution of minimum norm to the system of linear inequalities. This problem is equivalent to the linear least squares problem on the positive orthant. It is solved using orthogonal transformations, which are memorized as products. Like in the revised simplex method, an auxiliary matrix is used for computations. Compared to the modified-simplex type methods, the presented dual algorithm QPLS requires less storage and solves ill-conditioned problems more precisely. The algorithm is illustrated by some difficult problems.

Data mining the MNC like internal co-opetition duality in a university context

The goal of the paper is to quantify the simultaneous competition and cooperation that takes plac... more The goal of the paper is to quantify the simultaneous competition and cooperation that takes place in organizations. As the concepts seem to be dichotomous opposites at first, the term internal coopetition duality is put forth. Parallels are drawn between coopetitive processes in big multinational corporations (MNCs) and these taking place in universities, also the structural solutions used in both are analyzed. Data mining is used while looking at how specializations inside the university are in competition for better students. We look at the profiles that students have and find natural divisions between the specializations, by applying graph theory and modularity algorithms for community detection. The competitive position of the specializations is evident in the average grades of the detected communities. The ratio of intercommunity ties to intracommunity ties (conductance) quantifies the cooperative stance, though, as students with similar profiles express linkages in the curricula; and the choices regarding career development undertaken become evident. Managerial implications discussed include the imperative for actively managing and financially rewarding the outcomes of the coopetitive duality.

Mathematical programming via the least-squares method

Central European Journal of Mathematics, Jul 24, 2010

The least-squares method is used to obtain a stable algorithm for a system of linear inequalities... more The least-squares method is used to obtain a stable algorithm for a system of linear inequalities as well as linear and nonlinear programming. For these problems the solution with minimal norm for a system of linear inequalities is found by solving the non-negative least-squares (NNLS) problem. Approximate and exact solutions of these problems are discussed. Attention is mainly paid to finding the initial solution to an LP problem. For this purpose an NNLS problem is formulated, enabling finding the initial solution to the primal or dual problem, which may turn out to be optimal. The presented methods are primarily suitable for ill-conditioned and degenerate problems, as well as for LP problems for which the initial solution is not known. The algorithms are illustrated using some test problems.

Linear inequalities via least squares

Proceedings of the Estonian Academy of Sciences, 2013

Application of Orthogonal Transformations in the Revised Simplex Method

Unlike the classical method, orthogonal transformations are used for computing more precise simpl... more Unlike the classical method, orthogonal transformations are used for computing more precise simplex tables. These transformations are stored as products. With these transformations the basic matrix is turned to the triangular form, which is used instead of the reciprocal matrix. The values of initial and dual variables are found from the triangular system. The performed algorithm MULT is appropriate for

Predicting student retention by comparing histograms of bootstrapping for Charnes-Cooper transformationlinear programming discriminant analysis

2013 Second International Conference on E-Learning and E-Technologies in Education (ICEEE), 2013

ABSTRACT The goal of the paper is to predict student retention by using linear discriminant analy... more ABSTRACT The goal of the paper is to predict student retention by using linear discriminant analysis with bootstrapping. The result (93%) provides accuracy superior to the bootstrapping of a comparative method, as well as to the non-bootstrapping variations. In order to perform discriminant analysis, we linearize a fractional programming method by using Charnes-Cooper transformation and apply linear programming, while the comparative approach uses deviation variables to tackle a similar multiple criteria optimization problem. We train the discriminatory hyperplane family and apply it to the testing set - thus arriving at a set of histograms. We analyze the histograms by using the simple mean - best for prediction - and a five-fold Kolmogorov-Smirnov test - best used for resources allocation, in order to act on the final results. Final results are the outcome of applying the hyperplane family on freshman data.