In the presem paper, some important characteristics of Fenchel-, Frechet-,Hademard-, and Gateaux-Subdifferentials are showed up, and properties of functions, especially. convexity of functions, are described by subdif...In the presem paper, some important characteristics of Fenchel-, Frechet-,Hademard-, and Gateaux-Subdifferentials are showed up, and properties of functions, especially. convexity of functions, are described by subdifferentials.展开更多
An equation concerning with the subdifferential of convex functionals defined in real Banach spaces and the metric projections to level sets is shown. The equation is compared with the resolvents of general monotone o...An equation concerning with the subdifferential of convex functionals defined in real Banach spaces and the metric projections to level sets is shown. The equation is compared with the resolvents of general monotone operators, and makes the geometric properties of differential equations expressed by subdifferentials clear. Hence, it can be expected to be useful in obtaining the steepest descents defined by the convex functionals in Banach spaces. Also, it gives a similar result to the Lagrange multiplier method under certain conditions.展开更多
Ioffe’s approximate subdifferentials are reviewed and some of his resultsare generalized.An extension of the calculus of the approximate subdifferentials forthe sums to any finite number of functions is provided alon...Ioffe’s approximate subdifferentials are reviewed and some of his resultsare generalized.An extension of the calculus of the approximate subdifferentials forthe sums to any finite number of functions is provided along with a generalizationof the Dubovitzkii-Milyutin theorem.The presentation also indicates some of thelimitations of nonsmooth analysis and optimization.Restriction to the class offunction which is suitable for most of the purposes in nonsmooth optimization issuggested.展开更多
In this work, we study some subdifferentials of the distance function to a nonempty nonconvex closed subset of a general Banach space. We relate them to the normal cone of the enlargements of the set which can be cons...In this work, we study some subdifferentials of the distance function to a nonempty nonconvex closed subset of a general Banach space. We relate them to the normal cone of the enlargements of the set which can be considered as regularizations of the set.展开更多
This study introduces a novel voice cloning framework driven by Mordukhovich Subdifferential Optimization(MSO)to address the complex multi-objective challenges of pathological speech synthesis in underresourced Lithua...This study introduces a novel voice cloning framework driven by Mordukhovich Subdifferential Optimization(MSO)to address the complex multi-objective challenges of pathological speech synthesis in underresourced Lithuanian language with unique phonemes not present in most pre-trained models.Unlike existing voice synthesis models that often optimize for a single objective or are restricted to major languages,our approach explicitly balances four competing criteria:speech naturalness,speaker similarity,computational efficiency,and adaptability to pathological voice patterns.We evaluate four model configurations combining Lithuanian and English encoders,synthesizers,and vocoders.The hybrid model(English encoder,Lithuanian synthesizer,English vocoder),optimized via MSO,achieved the highest Mean Opinion Score(MOS)of 4.3 and demonstrated superior intelligibility and speaker fidelity.The results confirm that MSO enables effective navigation of trade-offs in multilingual pathological voice cloning,offering a scalable path toward high-quality voice restoration in clinical speech applications.This work represents the first integration of Mordukhovich optimization into pathological TTS,setting a new benchmark for speech synthesis under clinical and linguistic constraints.展开更多
We study a nonlinear periodic problem driven by the p（t）-Laplacian and having a nonsmooth potential （hemivariational inequalities）. Using a variational method based on nonsmooth critical point theory for locally L...We study a nonlinear periodic problem driven by the p（t）-Laplacian and having a nonsmooth potential （hemivariational inequalities）. Using a variational method based on nonsmooth critical point theory for locally Lipschitz functions, we first prove the existence of at least two nontrivial solutions under the generalized subquadratic and then establish the existence of at least one nontrivial solution under the generalized superquadratic.展开更多
uv-decomposition method for solving a mathematical program with equilibrium constraints （MPEC） problem with linear complementarity constraints is presented. The problem is first converted into a nonlinear programmin...uv-decomposition method for solving a mathematical program with equilibrium constraints （MPEC） problem with linear complementarity constraints is presented. The problem is first converted into a nonlinear programming one. The structure of subdifferential a corresponding penalty function and results of its uv-decomposition are given. A conceptual algorithm for solving this problem with a superUnear convergence rate is then constructed in terms of the obtained results.展开更多
In this paper we prove the existence and uniqueness of a weak solution for a dynamic electo-viscoetastic problem that describes a contact between a body and a foundation. We assume the body is made from thermoviscoela...In this paper we prove the existence and uniqueness of a weak solution for a dynamic electo-viscoetastic problem that describes a contact between a body and a foundation. We assume the body is made from thermoviscoelastic material and consider nonmonotone boundary conditions for the contact. We use recent results from the theory of hemivariational inequalities and the fixed point theory.展开更多
We study a new class of elliptic variational-hemivariational inequalities arising in the modelling of contact problems for elastic ideally locking materials. The contact is described by the Signorini unilateral contac...We study a new class of elliptic variational-hemivariational inequalities arising in the modelling of contact problems for elastic ideally locking materials. The contact is described by the Signorini unilateral contact condition and the friction is modelled by the nonmonotone multivalued subdifferential condition which depends on the slip. The problem is governed by a nonlinear elasticity operator, the subdifferential of the indicator function of a convex set which describes the locking constraints and a nonconvex locally Lipschitz friction potential. The result on existence and uniqueness of solution to the inequality is shown. The proof is based on a surjectivity result for maximal monotone and pseudomonotone operators combined with the application of the Banach contraction principle.展开更多
For a general normed vector space,a special optimal value function called a maximal time function is considered.This covers the farthest distance function as a special case,and has a close relationship with the smalle...For a general normed vector space,a special optimal value function called a maximal time function is considered.This covers the farthest distance function as a special case,and has a close relationship with the smallest enclosing ball problem.Some properties of the maximal time function are proven,including the convexity,the lower semicontinuity,and the exact characterizations of its subdifferential formulas.展开更多
One parabolic p-Laplacian-like differential equation with mixed boundaries is au/at in the corresponding studies is replaced by a（au/at）, studied in this paper, where the item au/at which makes it more general. The...One parabolic p-Laplacian-like differential equation with mixed boundaries is au/at in the corresponding studies is replaced by a（au/at）, studied in this paper, where the item au/at which makes it more general. The sufficient condition of the existence and uniqueness of non-trivial solution in L2（O, T; L2 （Ω）） is presented by employing the techniques of splitting the boundary problems into operator equation. Compared to the corresponding work, the restrictions imposed on the equation are weaken and the proof technique is simplified. It can be regarded as the extension and complement of the previous work.展开更多
A semilinear elliptic equation with strong resonance at infinity and with a nonsmooth potential is studied. Using nonsmooth critical point theory and developing some abstract minimax principles which complement and ex...A semilinear elliptic equation with strong resonance at infinity and with a nonsmooth potential is studied. Using nonsmooth critical point theory and developing some abstract minimax principles which complement and extend results in the literature, two results on existence are obtained.展开更多
In this paper, a nonlinear hemivariational inequality of second order with a forcing term of subcritical growth is studied. Using techniques from multivalued analysis and the theory of nonlinear operators of monotone ...In this paper, a nonlinear hemivariational inequality of second order with a forcing term of subcritical growth is studied. Using techniques from multivalued analysis and the theory of nonlinear operators of monotone type, an existence theorem for the Dirichlet boundary value problem is proved.展开更多
Numerical methods for the solution of nonsmooth equations are studied. A new subdifferential for a locally Lipschitzian function is proposed. Based on this subdifferential, Newton methods for solving nonsmooth equatio...Numerical methods for the solution of nonsmooth equations are studied. A new subdifferential for a locally Lipschitzian function is proposed. Based on this subdifferential, Newton methods for solving nonsmooth equations are developed and their convergence is shown. Since this subdifferential is easy to be computed, the present Newton methods can be executed easily in some applications.展开更多
The minimization of nonconvext nondifferentiable functions that are compositions of maxrtype functions formed by nondifferentiable convex functions is dialcussed in this paper. It is closely related to practical engin...The minimization of nonconvext nondifferentiable functions that are compositions of maxrtype functions formed by nondifferentiable convex functions is dialcussed in this paper. It is closely related to practical engineering problems. By utilizing the globality of ε-subdifferential and the theory of quasidifferential, and by introducing a new scheme which selects several search directions and consider them simultaneously at each iteration, a minimizing algorithm is derived. It is simple in structure, implemelltable, numerically efficient and has global convergence. The shortcomings of the existing algorithms are thus overcome both in theory and in application.展开更多
Motivated to obtain the second critical point of a nonlinear differential equation, which is expressed by derivatives of convex functional defined on a Banach space, an estimate with is given to see the relation ...Motivated to obtain the second critical point of a nonlinear differential equation, which is expressed by derivatives of convex functional defined on a Banach space, an estimate with is given to see the relation between f<sup>-1</sup>(0) and g<sup>-1</sup>(0). And both the Fréchet differentiability and the continuity of Fréchet derivative of every convex functional defined on an open subset of a Banach space are shown.展开更多
Abstract Recently a, monotone generalized directional derixrative has been introduced for Lipschitz functions. This concept has been applied to represent and optimize nonsmooth functions. The second a.pplication resul...Abstract Recently a, monotone generalized directional derixrative has been introduced for Lipschitz functions. This concept has been applied to represent and optimize nonsmooth functions. The second a.pplication result,ed relevant for parallel computing, by allowing to define minimization algorithms with high degree of inherent parallelism. The paper presents first the theoretical background, namely the notions of monotone generalized directional derivative and monotone generalized subdifferential. Then it defines the tools for the procedures, that is a necessary optimality condition and a steel>est descent direction. Therefore the minimization algorithms are outlined. Successively the used architectures and the performed numerical expertence are described, by listing and commenting the t.ested functions and the obtained results.展开更多
In this paper, the existence theorem of the cone weak subdifferential of set valued mapping in locally convex topological vector space is proved. Received March 30,1998. 1991 MR Subject Classification: 4...In this paper, the existence theorem of the cone weak subdifferential of set valued mapping in locally convex topological vector space is proved. Received March 30,1998. 1991 MR Subject Classification: 47H17,90C29.展开更多
In this paper we prove the convergence of the approximate proximal method for DC functions proposed by Sun et al [6]. Our analysis also permits to treat the exact method. We then propose an interesting result in the c...In this paper we prove the convergence of the approximate proximal method for DC functions proposed by Sun et al [6]. Our analysis also permits to treat the exact method. We then propose an interesting result in the case where the second component of the DC function is differentiable and provide some computational experiences which proved the efficiency of our method.展开更多
The question of establishing measure theory for statistical convergence has been moving closer to center stage, since a kind of reasonable theory is not only fundamental for unifying various kinds of statistical conve...The question of establishing measure theory for statistical convergence has been moving closer to center stage, since a kind of reasonable theory is not only fundamental for unifying various kinds of statistical convergence, but also a bridge linking the studies of statistical convergence across measure theory, integration theory, probability and statistics. For this reason, this paper, in terms of subdifferential, first shows a representation theorem for all finitely additive probability measures defined on the σ-algebra of all subsets of N, and proves that every such measure can be uniquely decomposed into a convex combination of a countably additive probability measure and a statistical measure (i.e. a finitely additive probability measure μ with μ(k) = 0 for all singletons {k}). This paper also shows that classical statistical measures have many nice properties, such as: The set of all such measures endowed with the topology of point-wise convergence on forms a compact convex Hausdorff space; every classical statistical measure is of continuity type (hence, atomless), and every specific class of statistical measures fits a complementation minimax rule for every subset in N. Finally, this paper shows that every kind of statistical convergence can be unified in convergence of statistical measures.展开更多
文摘In the presem paper, some important characteristics of Fenchel-, Frechet-,Hademard-, and Gateaux-Subdifferentials are showed up, and properties of functions, especially. convexity of functions, are described by subdifferentials.
文摘An equation concerning with the subdifferential of convex functionals defined in real Banach spaces and the metric projections to level sets is shown. The equation is compared with the resolvents of general monotone operators, and makes the geometric properties of differential equations expressed by subdifferentials clear. Hence, it can be expected to be useful in obtaining the steepest descents defined by the convex functionals in Banach spaces. Also, it gives a similar result to the Lagrange multiplier method under certain conditions.
文摘Ioffe’s approximate subdifferentials are reviewed and some of his resultsare generalized.An extension of the calculus of the approximate subdifferentials forthe sums to any finite number of functions is provided along with a generalizationof the Dubovitzkii-Milyutin theorem.The presentation also indicates some of thelimitations of nonsmooth analysis and optimization.Restriction to the class offunction which is suitable for most of the purposes in nonsmooth optimization issuggested.
基金The visit was made possible by financial supports from the Research Council of Hong-Kongthe General Consulate of France
文摘In this work, we study some subdifferentials of the distance function to a nonempty nonconvex closed subset of a general Banach space. We relate them to the normal cone of the enlargements of the set which can be considered as regularizations of the set.
基金funding from the Research Council of Lithuania(LMTLT),agreement No.S-MIP-23-46.
文摘This study introduces a novel voice cloning framework driven by Mordukhovich Subdifferential Optimization(MSO)to address the complex multi-objective challenges of pathological speech synthesis in underresourced Lithuanian language with unique phonemes not present in most pre-trained models.Unlike existing voice synthesis models that often optimize for a single objective or are restricted to major languages,our approach explicitly balances four competing criteria:speech naturalness,speaker similarity,computational efficiency,and adaptability to pathological voice patterns.We evaluate four model configurations combining Lithuanian and English encoders,synthesizers,and vocoders.The hybrid model(English encoder,Lithuanian synthesizer,English vocoder),optimized via MSO,achieved the highest Mean Opinion Score(MOS)of 4.3 and demonstrated superior intelligibility and speaker fidelity.The results confirm that MSO enables effective navigation of trade-offs in multilingual pathological voice cloning,offering a scalable path toward high-quality voice restoration in clinical speech applications.This work represents the first integration of Mordukhovich optimization into pathological TTS,setting a new benchmark for speech synthesis under clinical and linguistic constraints.
基金supported by the National Science Foundation of China (11001063, 10971043)the Fundamental Research Funds for the Central Universities (HEUCF 20111134)+2 种基金China Postdoctoral Science Foundation Funded Project (20110491032)Heilongjiang Provincial Science Foundation for Distinguished Young Scholars (JC200810)Program of Excellent Team in Harbin Institute of Technology and the Natural Science Foundation of Heilongjiang Province (A200803)
文摘We study a nonlinear periodic problem driven by the p（t）-Laplacian and having a nonsmooth potential （hemivariational inequalities）. Using a variational method based on nonsmooth critical point theory for locally Lipschitz functions, we first prove the existence of at least two nontrivial solutions under the generalized subquadratic and then establish the existence of at least one nontrivial solution under the generalized superquadratic.
基金Project supported by the National Natural Science Foundation of China(Nos.10372063,10771026 and 10471015)
文摘uv-decomposition method for solving a mathematical program with equilibrium constraints （MPEC） problem with linear complementarity constraints is presented. The problem is first converted into a nonlinear programming one. The structure of subdifferential a corresponding penalty function and results of its uv-decomposition are given. A conceptual algorithm for solving this problem with a superUnear convergence rate is then constructed in terms of the obtained results.
基金supported by the Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme under Grant Agreement No.295118the National Science Center of Poland under the Maestro Advanced Project No.DEC-2012/06/A/ST1/00262
文摘In this paper we prove the existence and uniqueness of a weak solution for a dynamic electo-viscoetastic problem that describes a contact between a body and a foundation. We assume the body is made from thermoviscoelastic material and consider nonmonotone boundary conditions for the contact. We use recent results from the theory of hemivariational inequalities and the fixed point theory.
基金supported by the National Science Center of Poland under the Maestro 3 Project No.DEC-2012/06/A/ST1/00262the project Polonium“Mathematical and Numerical Analysis for Contact Problems with Friction”2014/15 between the Jagiellonian University and Universitde Perpignan Via Domitia
文摘We study a new class of elliptic variational-hemivariational inequalities arising in the modelling of contact problems for elastic ideally locking materials. The contact is described by the Signorini unilateral contact condition and the friction is modelled by the nonmonotone multivalued subdifferential condition which depends on the slip. The problem is governed by a nonlinear elasticity operator, the subdifferential of the indicator function of a convex set which describes the locking constraints and a nonconvex locally Lipschitz friction potential. The result on existence and uniqueness of solution to the inequality is shown. The proof is based on a surjectivity result for maximal monotone and pseudomonotone operators combined with the application of the Banach contraction principle.
基金supported by the National Natural Science Foundation of China(11201324)the Fok Ying Tuny Education Foundation(141114)the Sichuan Technology Program(2022ZYD0011,2022NFSC1852).
文摘For a general normed vector space,a special optimal value function called a maximal time function is considered.This covers the farthest distance function as a special case,and has a close relationship with the smallest enclosing ball problem.Some properties of the maximal time function are proven,including the convexity,the lower semicontinuity,and the exact characterizations of its subdifferential formulas.
基金supported by the National Natural Science Foundation of China(11071053)Natural Science Foundation of Hebei Province(A2014207010)+1 种基金Key Project of Science and Research of Hebei Educational Department(ZD2016024)Key Project of Science and Research of Hebei University of Economics and Business(2015KYZ03)
文摘One parabolic p-Laplacian-like differential equation with mixed boundaries is au/at in the corresponding studies is replaced by a（au/at）, studied in this paper, where the item au/at which makes it more general. The sufficient condition of the existence and uniqueness of non-trivial solution in L2（O, T; L2 （Ω）） is presented by employing the techniques of splitting the boundary problems into operator equation. Compared to the corresponding work, the restrictions imposed on the equation are weaken and the proof technique is simplified. It can be regarded as the extension and complement of the previous work.
基金Research is supported by a grant of the National Scholarship Foundation of Greece (I.K.Y.)
文摘A semilinear elliptic equation with strong resonance at infinity and with a nonsmooth potential is studied. Using nonsmooth critical point theory and developing some abstract minimax principles which complement and extend results in the literature, two results on existence are obtained.
文摘In this paper, a nonlinear hemivariational inequality of second order with a forcing term of subcritical growth is studied. Using techniques from multivalued analysis and the theory of nonlinear operators of monotone type, an existence theorem for the Dirichlet boundary value problem is proved.
文摘Numerical methods for the solution of nonsmooth equations are studied. A new subdifferential for a locally Lipschitzian function is proposed. Based on this subdifferential, Newton methods for solving nonsmooth equations are developed and their convergence is shown. Since this subdifferential is easy to be computed, the present Newton methods can be executed easily in some applications.
文摘The minimization of nonconvext nondifferentiable functions that are compositions of maxrtype functions formed by nondifferentiable convex functions is dialcussed in this paper. It is closely related to practical engineering problems. By utilizing the globality of ε-subdifferential and the theory of quasidifferential, and by introducing a new scheme which selects several search directions and consider them simultaneously at each iteration, a minimizing algorithm is derived. It is simple in structure, implemelltable, numerically efficient and has global convergence. The shortcomings of the existing algorithms are thus overcome both in theory and in application.
文摘Motivated to obtain the second critical point of a nonlinear differential equation, which is expressed by derivatives of convex functional defined on a Banach space, an estimate with is given to see the relation between f<sup>-1</sup>(0) and g<sup>-1</sup>(0). And both the Fréchet differentiability and the continuity of Fréchet derivative of every convex functional defined on an open subset of a Banach space are shown.
文摘Abstract Recently a, monotone generalized directional derixrative has been introduced for Lipschitz functions. This concept has been applied to represent and optimize nonsmooth functions. The second a.pplication result,ed relevant for parallel computing, by allowing to define minimization algorithms with high degree of inherent parallelism. The paper presents first the theoretical background, namely the notions of monotone generalized directional derivative and monotone generalized subdifferential. Then it defines the tools for the procedures, that is a necessary optimality condition and a steel>est descent direction. Therefore the minimization algorithms are outlined. Successively the used architectures and the performed numerical expertence are described, by listing and commenting the t.ested functions and the obtained results.
文摘In this paper, the existence theorem of the cone weak subdifferential of set valued mapping in locally convex topological vector space is proved. Received March 30,1998. 1991 MR Subject Classification: 47H17,90C29.
文摘In this paper we prove the convergence of the approximate proximal method for DC functions proposed by Sun et al [6]. Our analysis also permits to treat the exact method. We then propose an interesting result in the case where the second component of the DC function is differentiable and provide some computational experiences which proved the efficiency of our method.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10771175, 10471114)
文摘The question of establishing measure theory for statistical convergence has been moving closer to center stage, since a kind of reasonable theory is not only fundamental for unifying various kinds of statistical convergence, but also a bridge linking the studies of statistical convergence across measure theory, integration theory, probability and statistics. For this reason, this paper, in terms of subdifferential, first shows a representation theorem for all finitely additive probability measures defined on the σ-algebra of all subsets of N, and proves that every such measure can be uniquely decomposed into a convex combination of a countably additive probability measure and a statistical measure (i.e. a finitely additive probability measure μ with μ(k) = 0 for all singletons {k}). This paper also shows that classical statistical measures have many nice properties, such as: The set of all such measures endowed with the topology of point-wise convergence on forms a compact convex Hausdorff space; every classical statistical measure is of continuity type (hence, atomless), and every specific class of statistical measures fits a complementation minimax rule for every subset in N. Finally, this paper shows that every kind of statistical convergence can be unified in convergence of statistical measures.
