Juan Parra - Academia.edu (original) (raw)

Papers by Juan Parra

Linear Algebra and Its Applications, 2002

In this paper we consider a parametrized family of linear inequality systems whose coefficients d... more In this paper we consider a parametrized family of linear inequality systems whose coefficients depend continuously on a parameter ranging in an arbitrary metric space. We analyze the lower semicontinuity (lsc) of the feasible set mapping in terms of the so-called carrier index set, consisting of those indices whose associated inequalities are satisfied as equalities at every feasible point. This concept, which leads to a weakened Slater condition, allows us to characterize the lsc of the feasible set mapping in terms of certain convex combinations of the coefficient vectors associated with the carrier indices. This property entails the lsc of the carrier feasible set mapping, assigning to each parameter the affine hull of the feasible set, which is also analyzed in this paper. The last section is concerned with the semi-infinite case. (M.A. López), canovas@umh.es (M.J. Cánovas), parra@ umh.es (J. Parra). 0024-3795/02/$ -see front matter 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 -3 7 9 5( 0 2) 0 0 3 7 1 -3

Siam Journal on Optimization, 2006

In this paper we measure how much a linear optimization problem, in R n , has to be perturbed in ... more In this paper we measure how much a linear optimization problem, in R n , has to be perturbed in order to loose either its solvability (i.e., the existence of optimal solutions) or its unsolvability property. In other words, if we consider as ill-posed problems those in the boundary of the set of solvable ones, this paper deals with the associated distance to ill-posedness. Our parameter space is the set of all the linear semiinfinite programming problems with a fixed, but arbitrary, index set. In this framework, which includes as a particular case the ordinary linear programming, we obtain a formula for the distance from a solvable problem to unsolvability in terms of the nominal problem's coefficients. Moreover, this formula also provides the exact expression, or a lower bound, of the distance from an unsolvable problem to solvability. 1 and practical applications, namely, stability of the feasible set ([2], [5], [17]), measures of conditioning ([9], [15]), complexity analysis of certain algorithms for computing solutions ([8], [10]), size of the feasible set ([2], [7]), and the optimal set, sensitivity of the optimal value, stability of the dual problem, metric regularity of mappings ([5], [6], [13]), etc.

Mathematical Programming, 2005

We obtain a formula for the modulus of metric regularity of a mapping defined by a semi-infinite ... more We obtain a formula for the modulus of metric regularity of a mapping defined by a semi-infinite system of equalities and inequalities. Based on this formula, we prove a theorem of Eckart-Young type for such set-valued infinite-dimensional mappings: given a metrically regular mapping F of this kind, the infimum of the norm of a linear function g such that F+g is not metrically regular is equal to the reciprocal to the modulus of regularity of F. The Lyusternik-Graves theorem gives a straightforward extension of these results to nonlinear systems. We also discuss the distance to infeasibility for homogeneous semi-infinite linear inequality systems.

Siam Journal on Optimization, 2007

This paper is concerned with the Lipschitzian behavior of the optimal set of convex semi-infinite... more This paper is concerned with the Lipschitzian behavior of the optimal set of convex semi-infinite optimization problems under continuous perturbations of the right hand side of the constraints and linear perturbations of the objective function. In this framework we provide a sufficient condition for the metric regularity of the inverse of the optimal set mapping. This condition consists of the Slater constraint qualification, together with a certain additional requirement in the Karush-Kuhn-Tucker conditions. For linear problems this sufficient condition turns out to be also necessary for the metric regularity, and it is equivalent to some well-known stability concepts.

Journal of Global Optimization, 2008

In this paper we make use of subdifferential calculus and other variational techniques, traced ou... more In this paper we make use of subdifferential calculus and other variational techniques, traced out from [Ioffe, A.D.: Metric regularity and subdifferential calculus. Uspekhi Mat. Nauk 55, 3(333), 103–162; Engligh translation Math. Surveys 55, 501–558 (2000); Ioffe, A.D.: On rubustness of the regularity property of maps. Control cybernet 32, 543–554 (2003)], to derive different expressions for the Lipschitz modulus of the optimal set mapping of canonically perturbed convex semi-infinite optimization problems. In order to apply this background for obtaining the modulus of metric regularity of the associated inverse multifunction, we have to analyze the stable behavior of this inverse mapping. In our semi-infinite framework this analysis entails some specific technical difficulties. We also provide a new expression of a global variational nature for the referred regularity modulus.

Siam Journal on Optimization, 1999

ABSTRACT This paper presents an approach to the stability and the Hadamard well-posedness of the ... more ABSTRACT This paper presents an approach to the stability and the Hadamard well-posedness of the linear semi-infinite programming problem (LSIP). No standard hypothesis is required in relation to the set indexing of the constraints and, consequently, the functional dependence between the linear constraints and their associated indices has no special property. We consider, as parameter space, the set of all LSIP problems whose constraint systems have the same index set, and we define in it an extended metric to measure the size of the perturbations. Throughout the paper the behavior of the optimal value function and of the optimal set mapping are analyzed. Moreover, a certain type of Hadamard well-posedness, which does not require the boundedness of the optimal set, is characterized. The main results provided in the paper allow us to point out that the lower semicontinuity of the feasible set mapping entails high stability of the whole problem, mainly when this property occurs simultaneously with the boundedness of the optimal set. In this case all the stability properties hold, with the only exception being the lower semicontinuity of the optimal set mapping.

Mathematics of Operations Research, 2006

Abstract We consider the parametric space of all the linear semi-infinite programming problems wi... more Abstract We consider the parametric space of all the linear semi-infinite programming problems with constraint systems having the same index set. Under a certain regularity condition, the so-called well-posedness with respect to the solvability, it is known from ...

Annals of Operations Research, 2001

In this paper we introduce the concept of solving strategy for a linear semi-infinite programming... more In this paper we introduce the concept of solving strategy for a linear semi-infinite programming problem, whose index set is arbitrary and whose coefficient functions have no special property at all. In particular, we consider two strategies which either approximately solve or exactly solve the approximating problems, respectively. Our principal aim is to establish a global framework to cope with different concepts of well-posedness spread out in the literature. Any concept of well-posedness should entail different properties of these strategies, even in the case that we are not assuming the boundedness of the optimal set. In the paper we consider three desirable properties, leading to an exhaustive study of them in relation to both strategies. The more significant results are summarized in a table, which allows us to show the double goal of the paper. On the one hand, we characterize the main features of each strategy, in terms of certain stability properties (lower and upper semicontinuity) of the feasible set mapping, optimal value function and optimal set mapping. On the other hand, and associated with some cells of the table, we recognize different notions of Hadamard well-posedness. We also provide an application to the analysis of the Hadamard well-posedness for a linear semi-infinite formulation of the Lagrangian dual of a nonlinear programming problem.

Mathematical Programming, 2005

In this paper we consider the parameter space of all the linear inequality systems, in the n-dime... more In this paper we consider the parameter space of all the linear inequality systems, in the n-dimensional Euclidean space, with a fixed and arbitrary (possibly infinite) index set. This parameter space is endowed with the topology of the uniform convergence of the coefficient vectors by means of an extended distance. Some authors, in a different context in which the index set is finite and, accordingly, the coefficients are bounded, consider the boundary of the set of consistent systems as the set of ill-posed systems. The distance from the nominal system to this boundary (‘distance to ill-posedness’), which constitutes itself a measure of the stability of the system, plays a decisive role in the complexity analysis of certain algorithms for finding a solution of the system. In our context, the presence of infinitely many constraints would lead us to consider separately two subsets of inconsistent systems, the so-called strongly inconsistent systems and the weakly inconsistent systems. Moreover, the possible unboundedness of the coefficient vectors of a system gives rise to a special subset formed by those systems whose distance to ill-posedness is infinite. Attending to these two facts, and according to the idea that a system is ill-posed when small changes in the system’s data yield different types of systems, now the boundary of the set of strongly inconsistent systems arises as the ‘generalized ill-posedness’ set. The paper characterizes this generalized ill-posedness of a system in terms of the so-called associated hypographical set, leading to an explicit formula for the ‘distance to generalized ill-posedness’. On the other hand, the consistency value of a system, also introduced in the paper, provides an alternative way to determine its distance to ill-posedness (in the original sense), and additionally allows us to distinguish the consistent well-posed systems from the inconsistent well-posed ones. The finite case is shown to be a meeting point of our linear semi-infinite approach to the distance to ill-posedness with certain results derived for conic linear systems. Applications to the analysis of the Lipschitz properties of the feasible set mapping, as well as to the complexity analysis of the ellipsoid algorithm, are also provided.

Mathematics of Operations Research, 2002

In this paper, we consider a parametric family of convex inequality systems in the Euclidean spac... more In this paper, we consider a parametric family of convex inequality systems in the Euclidean space, with an arbitrary infinite index set, T, and convex constraints depending continuously on a parameter ranging in a separable metric space. No structure is assumed for T, and so ...

Biomacromolecules, 2006

Polymers with eugenol moieties covalently bonded to the macromolecular chains were synthesized fo... more Polymers with eugenol moieties covalently bonded to the macromolecular chains were synthesized for potential application in orthopedic and dental cements. First, eugenol was functionalized with polymerizable groups. The synthetic methods employed afforded two different methacrylic derivatives, where the acrylic and eugenol moieties were either directly bonded, eugenyl methacrylate (EgMA), or separated through an oxyethylene group, ethoxyeugenyl methacrylate (EEgMA). A typical Fisher esterification reaction was used for the synthesis of EgMA and EEgMA, affording the desired monomers in 80% yields. Polymerization of each of the novel monomers, at low conversion, provided soluble polymers consisting of hydrocarbon macromolecules with pendant eugenol moieties. At high conversions only cross-linked polymers were obtained, attributed to participation of the allylic double bonds in the polymerization reaction. In addition, copolymers of each eugenol derivative with ethyl methacrylate (EMA) were prepared at low conversion, with the copolymerization reaction studied by assuming the terminal model and the reactivity ratios determined according to linear and nonlinear methods. The values obtained were r(EgMA) = 1.48, r(EMA) = 0.55 and r(EEgMA) = 1.22, r(EMA) = 0.42. High molecular weight polymers and copolymers were obtained at low conversion. Analysis of thermal properties revealed a T(g) of 95 degrees C for PEgMA and of 20 degrees C for PEEgMA and an increase in the thermal stability for the eugenol derivatives polymers and copolymers with respect to that of PEMA. Water sorption of the copolymers was found to decrease with the eugenol derivative content. Both monomers EgMA and EEgMA showed antibacterial activity against Streptococcus mutans, producing inhibition halos of 7 and 21 mm, respectively. Finally, cell culture studies revealed that the copolymers did not leach any toxic eluants and showed good cellular proliferation with respect to PEMA. This study thus indicates that the eugenyl methacrylate derivatives are potentially good candidates for dental and orthopedic cements.

Journal of Materials Science-materials in Medicine, 2007

Films and sponges of chitosan (CHI), chitosan/hyaluronic acid (CHI–HA) and chitosan/chondroitin s... more Films and sponges of chitosan (CHI), chitosan/hyaluronic acid (CHI–HA) and chitosan/chondroitin sulphate (CHI–CHOS), were prepared by film deposition or lyophilization (sponges), avoiding the formation of interpolyelectrolyte complexes. The biological behaviour of the systems was analysed by studying the cell behaviour using a fibroblast cell line and standard biological MTT and Alamar Blue tests. The morphology of films, sponges and cell seeded samples was analysed by ESEM. The results obtained indicate that all the systems can be considered as good supports for cell adhesion and proliferation, but there is specific activation of the proliferative process in the presence of hyaluronic acid and chondroitin sulphate.

Linear Algebra and Its Applications, 2002

In this paper we consider a parametrized family of linear inequality systems whose coefficients d... more In this paper we consider a parametrized family of linear inequality systems whose coefficients depend continuously on a parameter ranging in an arbitrary metric space. We analyze the lower semicontinuity (lsc) of the feasible set mapping in terms of the so-called carrier index set, consisting of those indices whose associated inequalities are satisfied as equalities at every feasible point. This concept, which leads to a weakened Slater condition, allows us to characterize the lsc of the feasible set mapping in terms of certain convex combinations of the coefficient vectors associated with the carrier indices. This property entails the lsc of the carrier feasible set mapping, assigning to each parameter the affine hull of the feasible set, which is also analyzed in this paper. The last section is concerned with the semi-infinite case. (M.A. López), canovas@umh.es (M.J. Cánovas), parra@ umh.es (J. Parra). 0024-3795/02/$ -see front matter 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 -3 7 9 5( 0 2) 0 0 3 7 1 -3

Siam Journal on Optimization, 2006

In this paper we measure how much a linear optimization problem, in R n , has to be perturbed in ... more In this paper we measure how much a linear optimization problem, in R n , has to be perturbed in order to loose either its solvability (i.e., the existence of optimal solutions) or its unsolvability property. In other words, if we consider as ill-posed problems those in the boundary of the set of solvable ones, this paper deals with the associated distance to ill-posedness. Our parameter space is the set of all the linear semiinfinite programming problems with a fixed, but arbitrary, index set. In this framework, which includes as a particular case the ordinary linear programming, we obtain a formula for the distance from a solvable problem to unsolvability in terms of the nominal problem's coefficients. Moreover, this formula also provides the exact expression, or a lower bound, of the distance from an unsolvable problem to solvability. 1 and practical applications, namely, stability of the feasible set ([2], [5], [17]), measures of conditioning ([9], [15]), complexity analysis of certain algorithms for computing solutions ([8], [10]), size of the feasible set ([2], [7]), and the optimal set, sensitivity of the optimal value, stability of the dual problem, metric regularity of mappings ([5], [6], [13]), etc.

Mathematical Programming, 2005

We obtain a formula for the modulus of metric regularity of a mapping defined by a semi-infinite ... more We obtain a formula for the modulus of metric regularity of a mapping defined by a semi-infinite system of equalities and inequalities. Based on this formula, we prove a theorem of Eckart-Young type for such set-valued infinite-dimensional mappings: given a metrically regular mapping F of this kind, the infimum of the norm of a linear function g such that F+g is not metrically regular is equal to the reciprocal to the modulus of regularity of F. The Lyusternik-Graves theorem gives a straightforward extension of these results to nonlinear systems. We also discuss the distance to infeasibility for homogeneous semi-infinite linear inequality systems.

Siam Journal on Optimization, 2007

This paper is concerned with the Lipschitzian behavior of the optimal set of convex semi-infinite... more This paper is concerned with the Lipschitzian behavior of the optimal set of convex semi-infinite optimization problems under continuous perturbations of the right hand side of the constraints and linear perturbations of the objective function. In this framework we provide a sufficient condition for the metric regularity of the inverse of the optimal set mapping. This condition consists of the Slater constraint qualification, together with a certain additional requirement in the Karush-Kuhn-Tucker conditions. For linear problems this sufficient condition turns out to be also necessary for the metric regularity, and it is equivalent to some well-known stability concepts.

Journal of Global Optimization, 2008

In this paper we make use of subdifferential calculus and other variational techniques, traced ou... more In this paper we make use of subdifferential calculus and other variational techniques, traced out from [Ioffe, A.D.: Metric regularity and subdifferential calculus. Uspekhi Mat. Nauk 55, 3(333), 103–162; Engligh translation Math. Surveys 55, 501–558 (2000); Ioffe, A.D.: On rubustness of the regularity property of maps. Control cybernet 32, 543–554 (2003)], to derive different expressions for the Lipschitz modulus of the optimal set mapping of canonically perturbed convex semi-infinite optimization problems. In order to apply this background for obtaining the modulus of metric regularity of the associated inverse multifunction, we have to analyze the stable behavior of this inverse mapping. In our semi-infinite framework this analysis entails some specific technical difficulties. We also provide a new expression of a global variational nature for the referred regularity modulus.

Siam Journal on Optimization, 1999

ABSTRACT This paper presents an approach to the stability and the Hadamard well-posedness of the ... more ABSTRACT This paper presents an approach to the stability and the Hadamard well-posedness of the linear semi-infinite programming problem (LSIP). No standard hypothesis is required in relation to the set indexing of the constraints and, consequently, the functional dependence between the linear constraints and their associated indices has no special property. We consider, as parameter space, the set of all LSIP problems whose constraint systems have the same index set, and we define in it an extended metric to measure the size of the perturbations. Throughout the paper the behavior of the optimal value function and of the optimal set mapping are analyzed. Moreover, a certain type of Hadamard well-posedness, which does not require the boundedness of the optimal set, is characterized. The main results provided in the paper allow us to point out that the lower semicontinuity of the feasible set mapping entails high stability of the whole problem, mainly when this property occurs simultaneously with the boundedness of the optimal set. In this case all the stability properties hold, with the only exception being the lower semicontinuity of the optimal set mapping.

Mathematics of Operations Research, 2006

Abstract We consider the parametric space of all the linear semi-infinite programming problems wi... more Abstract We consider the parametric space of all the linear semi-infinite programming problems with constraint systems having the same index set. Under a certain regularity condition, the so-called well-posedness with respect to the solvability, it is known from ...

Annals of Operations Research, 2001

In this paper we introduce the concept of solving strategy for a linear semi-infinite programming... more In this paper we introduce the concept of solving strategy for a linear semi-infinite programming problem, whose index set is arbitrary and whose coefficient functions have no special property at all. In particular, we consider two strategies which either approximately solve or exactly solve the approximating problems, respectively. Our principal aim is to establish a global framework to cope with different concepts of well-posedness spread out in the literature. Any concept of well-posedness should entail different properties of these strategies, even in the case that we are not assuming the boundedness of the optimal set. In the paper we consider three desirable properties, leading to an exhaustive study of them in relation to both strategies. The more significant results are summarized in a table, which allows us to show the double goal of the paper. On the one hand, we characterize the main features of each strategy, in terms of certain stability properties (lower and upper semicontinuity) of the feasible set mapping, optimal value function and optimal set mapping. On the other hand, and associated with some cells of the table, we recognize different notions of Hadamard well-posedness. We also provide an application to the analysis of the Hadamard well-posedness for a linear semi-infinite formulation of the Lagrangian dual of a nonlinear programming problem.

Mathematical Programming, 2005

In this paper we consider the parameter space of all the linear inequality systems, in the n-dime... more In this paper we consider the parameter space of all the linear inequality systems, in the n-dimensional Euclidean space, with a fixed and arbitrary (possibly infinite) index set. This parameter space is endowed with the topology of the uniform convergence of the coefficient vectors by means of an extended distance. Some authors, in a different context in which the index set is finite and, accordingly, the coefficients are bounded, consider the boundary of the set of consistent systems as the set of ill-posed systems. The distance from the nominal system to this boundary (‘distance to ill-posedness’), which constitutes itself a measure of the stability of the system, plays a decisive role in the complexity analysis of certain algorithms for finding a solution of the system. In our context, the presence of infinitely many constraints would lead us to consider separately two subsets of inconsistent systems, the so-called strongly inconsistent systems and the weakly inconsistent systems. Moreover, the possible unboundedness of the coefficient vectors of a system gives rise to a special subset formed by those systems whose distance to ill-posedness is infinite. Attending to these two facts, and according to the idea that a system is ill-posed when small changes in the system’s data yield different types of systems, now the boundary of the set of strongly inconsistent systems arises as the ‘generalized ill-posedness’ set. The paper characterizes this generalized ill-posedness of a system in terms of the so-called associated hypographical set, leading to an explicit formula for the ‘distance to generalized ill-posedness’. On the other hand, the consistency value of a system, also introduced in the paper, provides an alternative way to determine its distance to ill-posedness (in the original sense), and additionally allows us to distinguish the consistent well-posed systems from the inconsistent well-posed ones. The finite case is shown to be a meeting point of our linear semi-infinite approach to the distance to ill-posedness with certain results derived for conic linear systems. Applications to the analysis of the Lipschitz properties of the feasible set mapping, as well as to the complexity analysis of the ellipsoid algorithm, are also provided.

Mathematics of Operations Research, 2002

In this paper, we consider a parametric family of convex inequality systems in the Euclidean spac... more In this paper, we consider a parametric family of convex inequality systems in the Euclidean space, with an arbitrary infinite index set, T, and convex constraints depending continuously on a parameter ranging in a separable metric space. No structure is assumed for T, and so ...

Biomacromolecules, 2006

Polymers with eugenol moieties covalently bonded to the macromolecular chains were synthesized fo... more Polymers with eugenol moieties covalently bonded to the macromolecular chains were synthesized for potential application in orthopedic and dental cements. First, eugenol was functionalized with polymerizable groups. The synthetic methods employed afforded two different methacrylic derivatives, where the acrylic and eugenol moieties were either directly bonded, eugenyl methacrylate (EgMA), or separated through an oxyethylene group, ethoxyeugenyl methacrylate (EEgMA). A typical Fisher esterification reaction was used for the synthesis of EgMA and EEgMA, affording the desired monomers in 80% yields. Polymerization of each of the novel monomers, at low conversion, provided soluble polymers consisting of hydrocarbon macromolecules with pendant eugenol moieties. At high conversions only cross-linked polymers were obtained, attributed to participation of the allylic double bonds in the polymerization reaction. In addition, copolymers of each eugenol derivative with ethyl methacrylate (EMA) were prepared at low conversion, with the copolymerization reaction studied by assuming the terminal model and the reactivity ratios determined according to linear and nonlinear methods. The values obtained were r(EgMA) = 1.48, r(EMA) = 0.55 and r(EEgMA) = 1.22, r(EMA) = 0.42. High molecular weight polymers and copolymers were obtained at low conversion. Analysis of thermal properties revealed a T(g) of 95 degrees C for PEgMA and of 20 degrees C for PEEgMA and an increase in the thermal stability for the eugenol derivatives polymers and copolymers with respect to that of PEMA. Water sorption of the copolymers was found to decrease with the eugenol derivative content. Both monomers EgMA and EEgMA showed antibacterial activity against Streptococcus mutans, producing inhibition halos of 7 and 21 mm, respectively. Finally, cell culture studies revealed that the copolymers did not leach any toxic eluants and showed good cellular proliferation with respect to PEMA. This study thus indicates that the eugenyl methacrylate derivatives are potentially good candidates for dental and orthopedic cements.

Journal of Materials Science-materials in Medicine, 2007

Films and sponges of chitosan (CHI), chitosan/hyaluronic acid (CHI–HA) and chitosan/chondroitin s... more Films and sponges of chitosan (CHI), chitosan/hyaluronic acid (CHI–HA) and chitosan/chondroitin sulphate (CHI–CHOS), were prepared by film deposition or lyophilization (sponges), avoiding the formation of interpolyelectrolyte complexes. The biological behaviour of the systems was analysed by studying the cell behaviour using a fibroblast cell line and standard biological MTT and Alamar Blue tests. The morphology of films, sponges and cell seeded samples was analysed by ESEM. The results obtained indicate that all the systems can be considered as good supports for cell adhesion and proliferation, but there is specific activation of the proliferative process in the presence of hyaluronic acid and chondroitin sulphate.