Alternative Equivalent Formulation Research Articles

Algorithmic Aspect and Convergence Analysis for System of Generalized Multivalued Variational-like Inequalities

The main aim of this paper is twofold. Our first objective is to study a new system of generalized multivalued variational-like inequalities in Banach spaces and to establish its equivalence with a system of fixed point problems utilizing the concept of P-η-proximal mapping. The obtained alternative equivalent formulation is used and a new iterative algorithm for finding its approximate solution is suggested. Under some appropriate assumptions imposed on the mappings and parameters involved in the system of generalized multivalued variational-like inequalities, the existence of solution for the system mentioned above is proved and the convergence analysis of the sequences generated by our proposed iterative algorithm is discussed. The second objective of this work is to investigate and analyze the notion M-η-proximal mapping defined in the literature. Taking into account of the assumptions considered for such a mapping, we prove that every M-η-proximal mapping is actually P-η-proximal and is not a new one. At the same time, some comments relating to some existing results are pointed out.

MIXED QUASI VARIATIONAL INEQUALITIES INVOLVING FOUR NONLINEAR OPERATORS

In this paper we introduce and consider a new class of variational inequalities with four operators. This class is called the extended general mixed quasi variational inequality. We show that the extended general mixed quasi variational inequality is equivalent to the mixed point problem. We use this alternative equivalent formulation to discuss the existence of a solution of extended general mixed quasi variational inequality and also develop several iterative methods for solving extended general mixed quasi variational inequality and its variant forms. We consider the convergence analysis of the proposed iterative methods under appropriate conditions. We also introduce a new class of resolvent equation, which is called the extended general implicit resolvent equation and establish an equivalent relation between the extended general implicit resolvent equation and the extended general mixed quasi variational inequality. Some special cases are also discussed.

The onset of thermo-compositional convection in rotating spherical shells

ABSTRACTDouble-diffusive convection driven by both thermal and compositional buoyancy in a rotating spherical shell can exhibit a rather large number of behaviours often distinct from that of the single diffusive system. In order to understand how the differences in thermal and compositional molecular diffusivities determine the dynamics of thermo-compositional convection we investigate numerically the linear onset of convective instability in a double-diffusive setup. We construct an alternative equivalent formulation of the non-dimensional equations where the linearised double-diffusive problem is described by an effective Rayleigh number, , measuring the amplitude of the combined buoyancy driving, and a second parameter, α, measuring the mixing of the thermal and compositional contributions. This formulation is useful in that it allows for the analysis of several limiting cases and reveals dynamical similarities in the parameters space which are not obvious otherwise. We analyse the structure of the critical curves in this space, explaining asymptotic behaviours in α, transitions between inertial and diffusive regimes, and transitions between large-scale (fast drift) and small-scale (slow drift) convection. We perform this analysis for a variety of diffusivities, rotation rates and shell aspect ratios showing where and when new modes of convection take place.

Parallel schemes for solving a system of extended general quasi variational inequalities

In this paper, we consider a new system of extended general quasi variational inequalities involving six nonlinear operators. Using projection operator technique, we show that the system of extended general quasi variational inequalities is equivalent to a system of fixed point problems. Using this alternative equivalent formulation, we propose and analyze some parallel schemes for solving a system of extended general quasi variational inequalities. The convergence of these new schemes is discussed under some mild conditions. Several special cases are discussed. Results obtained in this paper continue to hold for these problems. The ideas and techniques of this paper may stimulate further research in this dynamic field.

General variational inclusions involving difference of operators

In this paper, we introduce and consider a new class of general variational inclusions involving the difference of operators in a Hilbert space. We establish the equivalence between the general variational inclusions and the fixed point problems as well as with a new class of resolvent equations using the resolvent operator technique. We use this alternative formulation to discuss the existence of a solution of the general variational inclusions. We again use this alternative equivalent formulation to suggest and analyze a number of iterative methods for finding a zero of the difference of operators. We also discuss the convergence of the iterative method under suitable conditions. Our methods of proofs are very simple as compared with other techniques. Several special cases of these problems are also considered. The results proved in this paper may be viewed as a refinement and an improvement of the known results in this area.

Some New Classes of Quasi Split Feasibility Problems

In thispaper, we introduce and consider a new problem of finding u2 K(u) such that Au2 C, where K : u! K(u) is a closed convex-valued set in the real Hilbert space H1, C is closed convex set in the real Hilbert space H2 respectively and A is linear bounded self-adjoint operator from H1 and H2. This problem is called the quasi split feasibility problem. We show that the qu asi feasibility problem is equivalent to the fixed point problem and quasi variational ine quality. These s alternative equivalent formulations are used to consider the existence of a solution of the quasi split feasibility problem. So me special cases are also considered. Problems considered in this paper may open further research opportunities in these fields.

Sensitivity analysis of general nonconvex variational inequalities

In this paper, we show that the parametric general nonconvex variational inequalities are equivalent to the parametric Wiener-Hopf equations. We use this alternative equivalent formulation to study the sensitivity analysis for the nonconvex variational inequalities without assuming the differentiability of the given data. Our results can be considered as a significant extension of previously known results for the variational inequalities.

Canonical and phantom scalar fields as an interaction of two perfect fluids

In this article we investigate and develop specific aspects of Friedmann-Robertson-Walker (FRW) scalar field cosmologies related to the interpretation that canonical and phantom scalar field sources may be interpreted as cosmological configurations with a mixture of two interacting barotropic perfect fluids: a matter component $\rho_1(t)$ with a stiff equation of state ($p_1=\rho_1$), and an "effective vacuum energy" $\rho_2(t)$ with a cosmological constant equation of state ($p_2=-\rho_2$). An important characteristic of this alternative equivalent formulation in the framework of interacting cosmologies is that it gives, by choosing a suitable form of the interacting term $Q$, an approach for obtaining exact and numerical solutions. The choice of $Q$ merely determines a specific scalar field with its potential, thus allowing to generate closed, open and flat FRW scalar field cosmologies.

On a system of extended general variational inclusions

At the present paper, a new system of extended general nonlinear variational inclusions is introduced and using the resolvent operator technique, equivalence between the aforesaid system and the fixed point problem is verified. By using this alternative equivalent formulation, the existence and uniqueness theorem for solution of the system of extended general nonlinear variational inclusions is demonstrated and two new iterative schemes for solving this system of extended general nonlinear variational inclusions are suggested and analyzed. The convergence analysis of the proposed iterative methods under some suitable conditions is studied. Some errors in a recent article by Noor et al. (J Inequal Appl 2011:10, 2011) are found and the incorrectness of the results of the cited paper is proved. Also, the results of the aforementioned paper are revised and corrected.

Flow constrained minimum cost flow problem

The paper deals with the uncapacitated minimum cost flow problems subject to additional flow constraints whether or not the sum of node capacities is zero. This is a generalization and extension of transportation problems with restrictions on total flow value. The relationship between the desired flow value and the sum of node capacities of source(s) and sink(s) gives rise to the different set of problems. Mathematical models for the various cases are formulated. For each case an equivalent standard minimum cost flow problem (MCFP) is formulated whose optimal solution provides the optimal solution to the original flow constrained problem. The paper not only extends the similar concept of transportation problem to the case of MCFP but also suggests an alternative equivalent formulation. Solving the alternative problem is computationally better in comparison to the formulation analogous to the transportation case. A broad computational comparison is carried out at the end.

Generalized AOR Method for Solving Absolute Complementarity Problems

We introduce and consider a new class of complementarity problems, which is called the absolute value complementarity problem. We establish the equivalence between the absolute complementarity problems and the fixed point problem using the projection operator. This alternative equivalent formulation is used to discuss the existence of a solution of the absolute value complementarity problem. A generalized AOR method is suggested and analyzed for solving the absolute the complementarity problems. We discuss the convergence of generalized AOR method for the L‐matrix. Several examples are given to illustrate the implementation and efficiency of the method. Results are very encouraging and may stimulate further research in this direction.

Parametric Extended General Mixed Variational Inequalities

It is well known that the resolvent equations are equivalent to the extended general mixed variational inequalities. We use this alternative equivalent formulation to study the sensitivity of the extended general mixed variational inequalities without assuming the differentiability of the given data. Since the extended general mixed variational inequalities include extended general variational inequalities, quasi (mixed) variational inequalities and complementarity problems as special cases, results obtained in this paper continue to hold for these problems. In fact, our results can be considered as a significant extension of previously known results.

Some Proximal Methods for Solving Mixed Variational Inequalities

It is well known that the mixed variational inequalities are equivalent to the fixed point problem. We use this alternative equivalent formulation to suggest some new proximal point methods for solving the mixed variational inequalities. These new methods include the explicit, the implicit, and the extragradient method as special cases. The convergence analysis of these new methods is considered under some suitable conditions. Our method of constructing these iterative methods is very simple. Results proved in this paper may stimulate further research in this direction.

On New Proximal Point Methods for Solving the Variational Inequalities

It is well known that the variational inequalities are equivalent to the fixed point problem. We use this alternative equivalent formulation to suggest and analyze some new proximal point methods for solving the variational inequalities. These new methods include the explicit, the implicit, and the extragradient methods as special cases. The convergence analysis of the new methods is considered under some suitable conditions. Results proved in this paper may stimulate further research in this direction.

Iterative methods for solving extended general mixed variational inequalities

In this paper, we introduce and consider a new class of mixed variational inequalities involving four operators, which are called extended general mixed variational inequalities. Using the resolvent operator technique, we establish the equivalence between the extended general mixed variational inequalities and fixed point problems as well as resolvent equations. We use this alternative equivalent formulation to suggest and analyze some iterative methods for solving general mixed variational inequalities. We study the convergence criteria for the suggested iterative methods under suitable conditions. Our methods of proof are very simple as compared with other techniques. The results proved in this paper may be viewed as refinements and important generalizations of the previous known results.

Projection iterative methods for solving some systems of general nonconvex variational inequalities

In this article, we introduce and consider a new system of general nonconvex variational inequalities involving four different operators. We use the projection operator technique to establish the equivalence between the system of general nonconvex variational inequalities and the fixed points problem. This alternative equivalent formulation is used to suggest and analyse some new explicit iterative methods for this system of nonconvex variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Since this new system includes the system of nonconvex variational inequalities, variational inequalities and related optimization problems as special cases, results obtained in this article continue to hold for these problems. Our results can be viewed as a refinement and an improvement of the previously known results for variational inequalities.

Combining initialization and solution inverse methods for inverse electrocardiography

Many methods have been developed over the last 25 years for the inverse problem of electrocardiography. The 2 main classes of such methods that have been of recent interest use 2 alternative equivalent source formulations. Activation-based methods were use as a source model a dipole layer on the heart surface with identical known transmembrane potential waveforms at each node. Potential-based methods use as a source model potential values at each time instant at each node on a closed surface surrounding the sources. The former is highly constrained, nonlinear, and nonconvex, whereas the latter is loosely constrained but linear. Recent interest in activation-based iterative solutions has focused on initialization. Much recent interest in potential-based solutions has focused on devising constraints, which impose physiologically meaningful waveform shapes. Here we describe work using one method to initialize a second for both formulations. For activation-based solutions, we initialize by embedding the solution set for the nonconvex optimization problem, which results from fixing waveforms and allowing only activation sample times to vary, into a less constrained solution set, thus formulating a convex problem. To do so, we replace the hard constraints on waveform shape by softer constraints. We solve this convex optimization problem combining constraints on the residual with the waveform shape constraints and imposing smoothness of the activation surface. The resulting optimal solution will not be a feasible set of activation waveforms, but we use it to obtain a good initialization for subsequent Newton-type iterations for the nonlinear, nonconvex problem. For the potential-based problem, we use a recent method from our group, Wavefront-Based Potential Reconstruction. Wavefront-Based Potential Reconstruction uses a constraint based on segmenting the reconstructed potentials from a previous iteration or time step into 3 spatial regions: activated, not yet activated, and transition. The first 2 are modeled as spatially constant, the third by a specified transition function. This constraint is used in a Tikhonov-like regularization, producing wavefronts without the typical regularized smoothing. Wavefront-Based Potential Reconstruction requires an initialization to remove reference drift to create the 3-region segmentation. We use a different potential-based method, the “isotropy method” from Huiskamp and Greensite, to estimate the required initialization parameters. In both cases, the combination of these 2 inversemethods shows, in simulations based on measured canine heart surface data and data from the ECGSim software package, improved results compared with standard approaches.

A resolvent method for solving mixed variational inequalities

It is well known that the mixed variational inequalities involving the nonlinear term are equivalent to the fixed-point problems. In this paper, we use this alternative equivalent formulation to suggest and analyze a new resolvent-type method for solving mixed variational inequalities. Our results can be viewed as significant extensions of the previously known results for mixed variational inequalities. An example is given to illustrate the efficiency and implementation of the proposed method.

On general quasi-variational inequalities

A new class of general quasi-variational inequalities involving two operators is introduced and studied. Using essentially the projection operator technique, we establish the equivalence between the general quasi-variational inequalities and the fixed-point problem and the Wiener–Hopf equations. These alternative equivalent formulations have been used to suggest and analyze several iterative methods for solving the general quasi-variational inequalities. We also discuss the convergence criteria of these iterative methods under some suitable conditions. Several special cases are also discussed.

Outer approximation algorithms for DC programs and beyond

This paper is a summary of the author’s Ph.D. thesis in Mathematics supervised by Giancarlo Bigi and Antonio Frangioni, and defended on 23 June 2008 at the Universita di Pisa. The thesis is written in English and available from http://www.di.unipi.it/di/groups/optimize. The work deals with outer approximation algorithms for solving an alternative equivalent formulation and a novel generalization of canonical DC (difference of convex) programs. The thesis develops an in-depth investigation of structural properties of both problems, and of the impact of approximations onto the quality of approximate optimal solutions. It also propose hierarchies of conditions guaranteeing convergence and develops several different implementable algorithms for canonical DC programs and its novel generalization, respectively.