Publications 2020

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29.  Mathematics Exercise Generator: the language of parameterized exercises

Oliveira, Paula and Carvalho, Paula

Developing Technology Mediation in Learning Environments

IGI Global

Nowadays, the process of teaching and learning is changing from a traditional model in which teachers were the source of information for a model in which teachers appear as advisors who carefully observe students, assist in the selection of information by identifying their learning needs and support students in their autonomous study. In this chapter, the authors describe an approach used in curricular units of first year in Science and Engineer degrees, which results from a connection of three projects born in University of Aveiro: MEGUA, SIACUA and PmatE, and the interconnections of their informatics platforms. Although any scientific area besides mathematics can use this tool, the authors focus in a case study using an example on a specific topic of Calculus courses for first year students on Engineering: Sequences and Series of Functions. The methodology described allows teachers to achieve further goals on learning strategies and students to have enough material to practice.

ria.ua.pt | Peer Reviewed

28.  Comparing learning objects for effective learning in Mathematics

Descalço, L. and Carvalho, P.

EDULEARN20 Proceedings

IATED

The too high number of students per class, situation typically found in higher education in Portugal, makes it difficult to adopt an approach were the student work is taken as the center of the learning process. The use of digital contents in this context is helpful for the students to build their own appropriate environment outside the classroom, using all support and learning material built up for that purpose. A fundamental problem we have in Mathematics is that many students are not well prepared in previous concepts. We have been developing and using several kinds of learning objects (LO) for calculus in the recent years, some of them dedicated to recall previous concepts. Among all kinds of contents, students valorize more learning objects that simultaneously challenge them and give them useful feedback. In this paper, we compare the usefulness of different kinds of LO, using computers, in a calculus course for students from several sciences and engineering, with a population of 290 students, based on a survey answered by 40% of them. The learning objects include multiple-choice questions with detailed solutions, quizzes, formative tests, short videos, and others. We present a quantitative analysis comparing different kinds of LO, considering students of several study areas. Together with the quantitative analysis, we add some qualitative considerations based on student’s answers to open questions concerning the learning methodology adopted.

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Articles


27.  Inventory models with reverse logistics for assets acquisition in a liquefied petroleum gas company

Lopes, Cristina and Correia, Aldina and Silva, Eliana Costa e and Monteiro, Magda and Lopes, Rui Borges

Journal of Mathematics in Industry

SpringerOpen

This paper addresses a case study regarding inventory models for acquiring liquefied petroleum gas (LPG) cylinders. This is an industrial challenge that was proposed at an European Study Group with Industry, by a Portuguese energy company, for which the LPG cylinder is the main asset of its LPG business. Due to the importance of this asset, an acquisition plan must be defined in order to determine the amount of LPG cylinders to acquire, and when to acquire them, in order to optimize the investment. As cylinders are returned and refilled, the reverse logistic flows of these assets must be considered. As the classical inventory models are not suitable for this case study, three new inventory models, which account for the return of LPG cylinders, are proposed in this work. The first proposed model considers deterministic constant demand and continuous returns of LPG cylinders, with discrete replenishment from the supplier. The second model is similar, but for the case when the returned cylinders cover for the demand. A third model is also proposed considering that both the demand and the returns are stochastic in nature and the replenishment from the supplier is discrete. The three models address different scenarios that the company is either currently facing or is expecting to occur in the near future.

ria.ua.pt | doi | Peer Reviewed

26.  Algorithmic determination of immobile indices in convex SIP problems with polyhedral index sets

Kostyukova, O. I. and Tchemisova, T. V.

INFOR: Information Systems and Operational Research

Taylor & Francis

The concepts of immobile indices and their immobility orders are objective and important characteristics of feasible sets of semi-infinite programming (SIP) problems. They can be used for the formulation of new efficient optimality conditions without constraint qualifications. Given a class of convex SIP problems with polyhedral index sets, we describe and justify a finite constructive algorithm (algorithm DIIPS) that allows to find in a finite number of steps all immobile indices and the corresponding immobility orders along the feasible directions. This algorithm is based on a representation of the cones of feasible directions in the polyhedral index sets in the form of linear combinations of extremal rays and on the approach proposed in our previous papers for the cases of immobile indices’ sets of simpler structures. A constructive procedure of determination of the extremal rays is described, and an example illustrating the application of the DIIPS algorithm is provided.

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25.  Extremal graphs for Estrada indices

Andrade, Enide and Lenes, Eber and Mallea-Zepeda, Exequiel and Robbiano, María and Rodríguez Z., Jonnathan

Linear Algebra and its Applications

Elsevier

Let $mathcal{G}$ be a simple undirected connected graph. The signless Laplacian Estrada, Laplacian Estrada and Estrada indices of a graph $mathcal{G}$ is the sum of the exponentials of the signless Laplacian eigenvalues, Laplacian eigenvalues and eigenvalues of $mathcal{G}$, respectively. The present work derives an upper bound for the Estrada index of a graph as a function of its chromatic number, in the family of graphs whose color classes have order not less than a fixed positive integer. The graphs for which the upper bound is tight is obtained. Additionally, an upper bound for the Estrada Index of the complement of a graph in the previous family of graphs with two color classes is given. A Nordhaus-Gaddum type inequality for the Laplacian Estrada index when {$mathcal{G}$ is a bipartite} graph with color classes of order not less than $2$, is presented. Moreover, a sharp upper bound for the Estrada index of the line graph and for the signless Laplacian index of a graph in terms of connectivity is obtained.

ria.ua.pt | doi | Peer Reviewed

24.  On the energy of singular and non singular graphs

Andrade, Enide and Carmona, Juan R. and Poveda, Alex and Robbiano, María

MATCH Communications in Mathematical and in Computer Chemistry

University of Kragujevac

Let $G$ be a simple undirected graph with $n$ vertices, $m$ edges, adjacency matrix $A$, largest eigenvalue $rho$ and nullity $kappa$. The energy of $G,$ $mathcal{E}(G)$ is the sum of its singular values. In this work lower bounds for $mathcal{E}(G)$ in terms of the coefficient of $mu^{kappa}$ in the expansion of characteristic polynomial, $p(mu)=det{(mu I-A)}$ are obtained. In particular one of the bounds generalizes a lower bound obtained by K. Das, S. A. Mojallal and I. Gutman in $2013$ to the case of graphs with given nullity. The bipartite case is also studied obtaining in this case, a sufficient condition to improve the spectral lower bound $2rho.$ Considering an increasing sequence convergent to $rho$ a convergent increasing sequence of lower bounds for the energy of $G$ is constructed.

ria.ua.pt | doi | Peer Reviewed

23.  A lower bound for the energy of hypoenergetic and non hypoenergetic graphs

Andrade, Enide and R. Carmona, Juan and Infante, Geraldine and Robbiano, María

MATCH Communications in Mathematical and in Computer Chemistry

University of Kragujevac, Faculty of Science

Let $G$ be a simple undirected graph with $n$ vertices and $m$ edges. The energy of $G,$ $mathcal{E}(G)$ corresponds to the sum of its singular values. This work obtains lower bounds for $mathcal{E}(G)$ where one of them generalizes a lower bound obtained by Mc Clelland in $1971$ to the case of graphs with given nullity. An extension to the bipartite case is given and, in this case, it is shown that the lower bound $2sqrt{m}$ is improved. The equality cases are characterized. Moreover, a simple lower bound that considers the number of edges and the diameter of $G$ is derived. A simple lower bound, which improves the lower bound $2sqrt{n-1}$, for the energy of trees with $n$ vertices and diameter $d$ is also obtained.

ria.ua.pt | Peer Reviewed

22.  Some new aspects of main eigenvalues of graphs

Abreu, Nair and Cardoso, Domingos M. and França, Francisca A. M. and Vinagre, Cybele T. M.

Computational and Applied Mathematics

Springer

An eigenvalue of the adjacency matrix of a graph is said to be main if the all-1 vector is non-orthogonal to the associated eigenspace. This paper explores some new aspects of the study of main eigenvalues of graphs, investigating specifically cones over strongly regular graphs and graphs for which the least eigenvalue is non-main. In this case, we characterize paths and trees with diameter-3 satisfying the property. We may note that the importance of least eigenvalues of graphs for the equilibria of social and economic networks was recently uncovered in literature.

ria.ua.pt | doi | Peer Reviewed

21.  Graphs with clusters perturbed by regular graphs: Aα-spectrum and applications

Cardoso, Domingos M. and Pastén, Germain and Rojo, Oscar

Discussiones Mathematicae Graph Theory

De Gruyter

Given a graph $G$, its adjacency matrix $A(G)$ and its diagonal matrix of vertex degrees $D(G)$, consider the matrix $A_{alpha}left( Gright) = alpha Dleft( Gright) +(1-alpha)Aleft(Gright)$, where $alpha inleft[ 0,1right)$. The $A_{alpha}-$ spectrum of $G$ is the multiset of eigenvalues of $A_{alpha}(G)$ and these eigenvalues are the $alpha-$ eigenvalues of $G$. A cluster in $G$ is a pair of vertex subsets $(C,S)$, where $C$ is a set of cardinality $|C| ge 2$ of pairwise co-neighbor vertices sharing the same set $S$ of $|S|$ neighbors. Assuming that $G$ is connected and it has a cluster $(C,S)$, $G(H)$ is obtained from $G$ and an $r-$ regular graph $H$ of order $|C|$ by identifying its vertices with the vertices in $C$, eigenvalues of $A_{alpha}(G)$ and $A_{alpha}(G(H))$ are deduced and if $A_{alpha}(H)$ is positive semidefinite then the $i$-th eigenvalue of $A_{alpha}(G(H))$ is greater than or equal to $i$-th eigenvalue of $A_{alpha}(G)$. These results are extended to graphs with several pairwise disjoint clusters $(C_1,S_1), ldots, (C_k,S_k)$. As an application, the effect on the energy, $alpha$-Estrada index and $alpha$-index of a graph $G$ with clusters when the edges of regular graphs are added to $G$ are analyzed. Finally, the $A_{alpha}-$ spectrum of the corona product $G circ H$ of a connected graph $G$ and a regular graph $H$ is determined.

ria.ua.pt | doi | Peer Reviewed

20.  A survey on graphs with convex quadratic stability number

Cardoso, Domingos M.

Optimization

Taylors & Fancis

A graph with convex quadratic stability number is a graph for which the stability number is determined by solving a convex quadratic program. Since the very beginning, where a convex quadratic programming upper bound on the stability number was introduced, a necessary and sufficient condition for this upper bound be attained was deduced. The recognition of graphs with convex quadratic stability number has been deeply studied with several consequences from continuous and combinatorial point of view. This survey starts with an exposition of some extensions of the classical Motzkin-Straus approach to the determination of the stability number of a graph and its relations with the convex quadratic programming upper bound. The main advances, including several properties and alternative characterizations of graphs with convex quadratic stability number are described as well as the algorithmic strategies developed for their recognition. Open problems and a conjecture for a particular class of graphs, herein called adverse graphs, are presented, pointing out a research line which is a challenge between continuous and discrete problems.

ria.ua.pt | doi | Peer Reviewed

19.  On the spectra of some g-circulant matrices and applications to nonnegative inverse eigenvalue problem

Andrade, Enide and Arrieta, Luis and Manzaneda, Cristina and Robbiano, María

Linear Algebra and its Applications

Elsevier

A $g$-circulant matrix $A$, is defined as a matrix of order $n$ where the elements of each row of $A$ are identical to those of the previous row, but are moved $g$ positions to the right and wrapped around. Using number theory, certain spectra of $g$-circulant real matrices are given explicitly. The obtained results are applied to Nonnegative Inverse Eigenvalue Problem to construct nonnegative, $g$-circulant matrices with given appropriated spectrum. Additionally, some $g$-circulant marices are reconstructed from its main diagonal entries.

ria.ua.pt | doi | Peer Reviewed

18.  An exploration of locally spherical regular hypertope

Fernandes, Maria Elisa and Leemans, Dimitri and Weiss, Asia Ivić

Discrete & Computational Geometry

Springer Nature

Hypertope is a generalization of the concept of polytope, which in turn generalizes the concept of a map and hypermap, to higher rank objects. Regular hypertopes with spherical residues, which we call regular locally spherical hypertopes, must be either of spherical, euclidean, or hyperbolic type. That is, the type-preserving automorphism group of a locally spherical regular hypertope is a quotient of a finite irreducible, infinite irreducible, or compact hyperbolic Coxeter group. We classify the locally spherical regular hypertopes of spherical and euclidean type and investigate finite hypertopes of hyperbolic type, giving new examples and summarizing some known results.

ria.ua.pt | doi | Peer Reviewed

17.  A multi-model methodology for forecasting sales and returns of liquefied petroleum gas cylinders

Correia, A. and Lopes, C. and Costa e Silva, E. and Monteiro, M. and Borges Lopes, R.

Neural Computing and Applications

Springer-Verlag London

In the liquefied petroleum gas (LPG) cylinder business, one of the most important assets is the LPG cylinder. This work addresses the asset acquisition planning for the LPG cylinder business of a company from the energy sector which has recently started this activity. In order to make the acquisition plan, it was necessary to forecast the sales and the LPG cylinder return rate. For that purpose, an ensemble method using time series techniques, multiple linear regression models and artificial neural networks was employed. Sales forecast was obtained using time series techniques, in particular, moving averages and exponential smoothing. Then, forecast of bottled propane gas sales and return rate was also addressed through multiple linear regression and artificial neural networks. A probability density function was defined for each of the different approaches. Afterward, using Monte Carlo simulation, the forecast values are obtained by a linear combination of the probability density functions, thus producing the final forecast. Results show that the company’s expectation of growth is larger than that predicted by the proposed methodology, which means the company should reflect on its current asset acquisition strategy. By combining different approaches, the proposed multi-model methodology allowed to obtain an accurate forecasting, without requiring a lot of historical data.

ria.ua.pt | doi | Peer Reviewed

16.  Spectral inequalities for Kubo-Ando operator means

Lemos, Rute and Soares, Graça

Linear Algebra and its Applications

Elsevier

Eigenvalue and singular value inequalities, involving Kubo-Ando operator connections or means, are established. Previous results from Lemos and Soares in 2018 are generalized or complemented, some log-majorizations are included. As a consequence, a refinement of an independent result by Ando and by Visick in 1995 on the eigenvalues of the Hadamard product is derived. Some singular value inequalities by Zou in 2017 are further extended.

ria.ua.pt | doi | Peer Reviewed

15.  Global solution of the initial value problem for the focusing Davey-Stewartson II system

Lakshtanov, Evgeny and Vainberg, Boris

Pure and Applied Functional Analysis

Yokohama Publishers

We consider the two dimensional focusing Davey-Stewartson II system and construct the global solution of the Cauchy problem for a dense in L 2 (C) set of initial data. We do not assume that the initial data is small. So, the solutions may have singularities. We show that the blow-up may occur only on a real analytic variety and the variety is bounded in each strip t ≤ T.

ria.ua.pt | Peer Reviewed

14.  Temporal constraints and device management for the Skill VRP: mathematical model and lower bounding techniques

Cappanera, Paola and Requejo, Cristina and Scutellà, Maria Grazia

Computers & Operations Research

Elsevier

We study a generalization of the Skill VRP that incorporates time windows aspects, precedence and synchronization constraints. Specifically, we are given a logistic network where nodes correspond to customers, and where each customer requires a set of (partially ordered) operations. A set of technicians is available to perform such operations, and each technician is qualified to execute only a subset of them, depending on his skill. By referring to a specific context such as Health Care, customers are patients while technicians are caregivers. In a Field Service context, instead, customers are usually referred to as clients while technicians as field technicians. The innovative aspect is that some operations may require a special device, which must be transported at the customer site and must be present at the customer location together with a technician qualified to use it. Given technician dependent traveling costs, we address the problem of defining the tours for the technicians and for the special device, while respecting the skill compatibility between customers and technicians, and the time windows, precedence and synchronization constraints. We propose a Mixed Integer Linear Programming (MILP) model for the generalized Skill VRP, and present some lower bounding techniques based on the proposed formulation. Preliminary computational experiments show that some lower bounding techniques may rapidly produce good lower bounds, thanks to quite effective valid inequalities. The returned percentage optimality gaps, estimated also thanks to a simple matheuristic, are in fact quite small for several scenarios of medium to large size, by encouraging the use of the proposed lower bounding techniques both as building blocks for designing exact approaches, and also as valuable tools to evaluate the efficacy of more sophisticated heuristic approaches to the problem.

ria.ua.pt | doi | Peer Reviewed

13.  Efficient lower and upper bounds for the weight-constrained minimum spanning tree problem using simple Lagrangian based algorithms

Requejo, Cristina and Santos, Eulália

Operational Research

Springer

The weight-constrained minimum spanning tree problem (WMST) is a combinatorial optimization problem for which simple but effective Lagrangian based algorithms have been used to compute lower and upper bounds. In this work we present several Lagrangian based algorithms for the WMST and propose two new algorithms, one incorporates cover inequalities. A uniform framework for deriving approximate solutions to the WMST is presented. We undertake an extensive computational experience comparing these Lagrangian based algorithms and show that these algorithms are fast and present small integrality gap values. The two proposed algorithms obtain good upper bounds and one of the proposed algorithms obtains the best lower bounds to the WMST.

ria.ua.pt | doi | Peer Reviewed

12.  Robust inventory theory with perishable products

Santos, Marcio Costa and Agra, Agostinho and Poss, Michael

Annals of Operations Research

Springer

We consider a robust inventory problem where products are perishable with a given shelf life and demands are assumed uncertain and can take any value in a given polytope. Interestingly, considering uncertain demands leads to part of the production being spoiled, a phenomenon that does not appear in the deterministic context. Based on a deterministic model we propose a robust model where the production decisions are first-stage variables and the inventory levels and the spoiled production are recourse variables that can be adjusted to the demand scenario following a FIFO policy. To handle the non-anticipativity constraints related to the FIFO policy, we propose a non-linear reformulation for the robust problem, which is then linearized using classical techniques. We propose a row-and-column generation algorithm to solve the reformulated model to optimality using a decomposition algorithm. Computational tests show that the decomposition approach can solve a set of instances representing different practical situations within reasonable amount of time. Moreover, the robust solutions obtained ensure low losses of production when the worst-case scenarios are materialized.

ria.ua.pt | doi | Peer Reviewed

11.  Heuristics for a vehicle routing problem with information collection in wireless networks

Flores-Luyo, Luis and Agra, Agostinho and Figueiredo, Rosa and Ocaña, Eladio

Journal of Heuristics

Springer

We consider a wireless network where a given set of stations is continuously generating information. A single vehicle, located at a base station, is available to collect the information via wireless transfer. The wireless transfer vehicle routing problem (WTVRP) is to decide which stations should be visited in the vehicle route, how long shall the vehicle stay in each station, and how much information shall be transferred from the nearby stations to the vehicle during each stay. The goal is to collect the maximum amount of information during a time period after which the vehicle returns to the base station. The WTVRP is NP-hard. Although it can be solved to optimality for small size instances, one needs to rely on good heuristic schemes to obtain good solutions for large size instances. In this work, we consider a mathematical formulation based on the vehicle visits. Several heuristics strategies are proposed, most of them based on the mathematical model. These strategies include constructive and improvement heuristics. Computational experiments show that a strategy that combines a combinatorial greedy heuristic to design a initial vehicle route, improved by a fix-and-optimize heuristic to provide a local optimum, followed by an exchange heuristic, affords good solutions within reasonable amount of running time.

ria.ua.pt | doi | Peer Reviewed

10.  A note on Newton's problem of minimal resistance for convex bodies

Plakhov, Alexander

Calculus of Variations and Partial Differential Equations

Springer

We consider the following problem: minimize the functional f (∇u(x)) dx in the class of concave functions u : D → [0, M], where D ⊂ R2 is a convex body and M > 0. If f (x) = 1/(1 + |x|^2) and D is a circle, the problem is called Newton’s problem of least resistance. It is known that the problem admits at least one solution. We prove that if all points of ∂D are regular and (1 + |x|) f (x)/(|y| f (y)) → +∞ as (1 + |x|)/|y| → 0 then a solution u to the problem satisfies u|_∂D = 0. This result proves the conjecture stated in 1993 in the paper by Buttazzo and Kawohl (Math Intell 15:7–12, 1993) for Newton’s problem.

ria.ua.pt | doi | Peer Reviewed

9.  Optimal impulse control of a SIR epidemic

Piunovskiy, Alexey and Plakhov, Alexander and Tumanov, Mikhail

Optimal Control Applications and Methods

John Wiley and Sons; Wiley

Based on our recent results on the optimal impulse control, we solve explicitly an optimal isolation problem for a specific SIR (susceptible-infective-removed) epidemic model describing, eg, the spread of AIDS.

ria.ua.pt | doi | Peer Reviewed

8.  Spectral properties of the n-Queens' graphs

Cardoso, Domingos M. and Costa, Inês Serôdio and Duarte, Rui

arXiv

The n-Queens’ graph, Q(n), is the graph associated to the n×n chessboard (a generalization of the classical 8×8 chessboard), with n 2 vertices, each one corresponding to a square of the chessboard. Two vertices of Q(n) are adjacent if and only if they are in the same row, in the same column or in the same diagonal of the chessboard. After a short overview on the main combinatorial properties of Q(n), its spectral properties are investigated. First, a lower bound on the least eigenvalue of an arbitrary graph is obtained using clique edge partitions and a sufficient condition for this lower bound be attained is deduced. For the particular case of Q(n), we prove that for every n, its least eigenvalue is not less than −4 and it is equal to −4 with multiplicity (n − 3)2 , for every n ≥ 4. Furthermore, n − 4 is also an eigenvalue of Q(n), with multiplicity at least n−2 2 when n is even and at least n+1 2 when n is odd. A conjecture about the integer eigenvalues of Q(n) is presented. We finish this article with an algorithm to determine an equitable partition of the n-Queens’ graph, Q(n), for n ≥ 3, concluding that such equitable partition has (⌈n/2⌉+1)⌈n/2⌉ 2 cells.

ria.ua.pt

7.  Immobile indices and CQ-free optimality criteria for linear copositive programming problems

Kostyukova, O. I. and Tchemisova, T. V. and Dudina, O. S.

Set-Valued and Variational Analysis

Springer

We consider problems of linear copositive programming where feasible sets consist of vectors for which the quadratic forms induced by the corresponding linear matrix combinations are nonnegative over the nonnegative orthant. Given a linear copositive problem, we define immobile indices of its constraints and a normalized immobile index set. We prove that the normalized immobile index set is either empty or can be represented as a union of a finite number of convex closed bounded polyhedra. We show that the study of the structure of this set and the connected properties of the feasible set permits to obtain new optimality criteria for copositive problems. These criteria do not require the fulfillment of any additional conditions (constraint qualifications or other). An illustrative example shows that the optimality conditions formulated in the paper permit to detect the optimality of feasible solutions for which the known sufficient optimality conditions are not able to do this. We apply the approach based on the notion of immobile indices to obtain new formulations of regularized primal and dual problems which are explicit and guarantee strong duality.

ria.ua.pt | doi | Peer Reviewed

6.  CQ-free optimality conditions and strong dual formulations for a special conic optimization problem

Kostyukova, Olga and Tchemisova, Tatiana

Statistics, Optimization & Information Computing

International Academic Press

In this paper, we consider a special class of conic optimization problems, consisting of set-semidefinite (orK-semidefinite) programming problems, where the setKis a polyhedral convex cone. For these problems, we introduce theconcept of immobile indices and study the properties of the set of normalized immobile indices and the feasible set. Thisstudy provides the main result of the paper, which is to formulate and prove the new first-order optimality conditions inthe form of a criterion. The optimality conditions are explicit and do not use any constraint qualifications. For the case of alinear cost function, we reformulate theK-semidefinite problem in a regularized form and construct its dual. We show thatthe pair of the primal and dual regularized problems satisfies the strong duality relation which means that the duality gap is vanishing.

ria.ua.pt | doi | Peer Reviewed

5.  Linear semidefinite programming problems: regularisation and strong dual formulations

Kostyukova, O. I. and Tchemisova, T. V.

Journal of the Belarusian State University: Mathematics and Informatics

Belarusian State University

Regularisation consists in reducing a given optimisation problem to an equivalent form where certain regularity conditions, which guarantee the strong duality, are fulfilled. In this paper, for linear problems of semidefinite programming (SDP), we propose a regularisation procedure which is based on the concept of an immobile index set and its properties. This procedure is described in the form of a finite algorithm which converts any linear semidefinite problem to a form that satisfies the Slater condition. Using the properties of the immobile indices and the described regularisation procedure, we obtained new dual SDP problems in implicit and explicit forms. It is proven that for the constructed dual problems and the original problem the strong duality property holds true.

ria.ua.pt | doi | Peer Reviewed

4.  On strong duality in linear copositive programming

Kostyukova, O. I. and Tchemisova, T. V.

arXiv

The paper is dedicated to the study of strong duality for a problem of linear copositive programming. Based on the recently introduced concept of the set of normalized immobile indices, an extended dual problem is deduced. The dual problem satisfies the strong duality relations and does not require any additional regularity assumptions such as constraint qualifications. The main difference with the previously obtained results consists in the fact that now the extended dual problem uses neither the immobile indices themselves nor the explicit information about the convex hull of these indices. The strong duality formulations presented in the paper have similar structure and properties as that proposed in the works of M. Ramana, L. Tuncel, and H. Wolkovicz, for semidefinite programming, but are obtained using different techniques.

ria.ua.pt

3.  On equivalent representations and properties of faces of the cone of copositive matrices

Kostyukova, O. I. and Tchemisova, T. V.

arXiv

The paper is devoted to a study of the cone COPp of copositive matrices. Based on the known from semi-infinite optimization concept of immobile indices, we define zero and minimal zero vectors of a subset of the cone COPp and use them to obtain different representations of faces of COPp and the corresponding dual cones. We describe the minimal face of COPp containing a given convex subset of this cone and prove some propositions that allow to obtain equivalent descriptions of the feasible sets of a copositive problems and may be useful for creating new numerical methods based on their regularization.

ria.ua.pt

2.  The main vertices of a star set and related graph parameters

Anđelić, Milica and Cardoso, Domingos M. and Simić, Slobodan K. and Stanić, Zoran

arXiv

A vertex v ∈ V (G) is called λ-main if it belongs to a star set X ⊂ V (G) of the eigenvalue λ of a graph G and this eigenvalue is main for the graph obtained from G by deleting all the vertices in X {v}; otherwise, v is λ-non-main. Some results concerning main and non-main vertices of an eigenvalue are deduced. For a main eigenvalue λ of a graph G, we introduce the minimum and maximum number of λ-main vertices in some λ-star set of G as new graph invariant parameters. The determination of these parameters is formulated as a combinatorial optimization problem based on a simplex-like approach. Using these and some related parameters we develop new spectral tools that can be used in the research of the isomorphism problem. Examples of graphs for which the maximum number of λ-main vertices coincides with the cardinality of a λ-star set are provided.

ria.ua.pt

1.  On the equivalence of the integral and differential Bellman equations in impulse control problems

Dufour, Francois and Piunovskiy, Alexey and Plakhov, Alexander

International Journal of Control

Taylor & Francis

When solving optimal impulse control problems, one can use the dynamic programming approach in two different ways: at each time moment, one can make the decision whether to apply a particular type of impulse, leading to the instantaneous change of the state, or apply no impulses at all; or, otherwise, one can plan an impulse after a certain interval, so that the length of that interval is to be optimized along with the type of that impulse. The first method leads to the differential Bellman equation, while the second method leads to the integral Bellman equation. The target of the current article is to prove the equivalence of those Bellman equations. Firstly, we prove that, for the simple deterministic optimal stopping problem, the equations in the integral and differential form are equivalent under very mild conditions. Here, the impulse means that the uncontrolled process is stopped, i.e., sent to the so called cemetery. After that, the obtained result immediately implies the similar equivalence of the Bellman equations for other models of optimal impulse control. Those include abstract dynamical systems, controlled ordinary differential equations, piece-wise deterministic Markov processes and continuous-time Markov decision processes.

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