Multiobjective Variational Problems Research Articles

Sufficient Efficiency Conditions for Vector Ratio Problem on the Second‐Order Jet Bundle

Motivated by its possible applications in mechanics and mechanical engineering, in our previous published work (Pitea and Postolache, 2011), we initiated an optimization theory for the second‐order jet bundle. We considered the problem of minimization of vectors of curvilinear functionals (well known as mechanical work), thought as multitime multiobjective variational problems, subject to PDE and/or PDI constraints. Within this framework, we introduced necessary optimality conditions. As natural continuation of these results, the present work introduces a study of sufficient efficiency conditions.

Minimization of Vectors of Curvilinear Functionals on Second‐Order Jet Bundle: Dual Program Theory

A previous paper (2011), Pitea and Postolache, considered the problem of minimization of vectors of curvilinear functionals (well known as mechanical work), thought as multitime multiobjective variational problem, subject to PDE and/or PDI constraints. They have chosen the suitable framework offered by the second‐order jet bundle, and initiated an optimization theory for this class of problems by introducing necessary conditions. As natural continuation of these results, the present work introduces a dual program theory, the general setting, and the theory which is new as a whole, containing our results.

Sufficiency and duality for multiobjective variational control problems with [formula omitted]-invexity

Antczak introduced vector-valued G-invex functions in 2009, which is a new class of generalized convex functions for differentiable multiobjective programming problems. In this paper, we extend the vector-valued G-invex functions to multiobjective variational control problems. By using the new concepts, a number of sufficient optimality results and Mond–Weir type duality results are obtained for multiobjective variational control programming problems.

Minimization of vectors of curvilinear functionals on the second order jet bundle: sufficient efficiency conditions

Strongly motivated by its possible applications in Mechanics, in our previous work (Pitea and Postolache (Optim. Lett. doi:10.1007/s11590-010-0272-0, 2011)), we initiated an optimization theory for the second order jet bundle. We considered the problem of minimization of vectors of curvilinear functionals (well known as mechanical work), thought as multi-time multi-objective variational problems, subject to PDE and/or PDI constraints. Within this framework, we introduced necessary conditions. As natural continuation of our results in Pitea and Postolache (Optim. Lett. doi:10.1007/s11590-010-0272-0, 2011), the present work introduces a study of sufficient efficiency conditions. While the background in Sect. 2 is introductory, the theory in Sect. 3 is new as a whole, containing our results.

Duality theorems for a new class of multitime multiobjective variational problems

In this work, we consider a new class of multitime multiobjective variational problems of minimizing a vector of functionals of curvilinear integral type. Based on the normal efficiency conditions for multitime multiobjective variational problems, we study duals of Mond-Weir type, generalized Mond-Weir-Zalmai type and under some assumptions of (?, b)-quasiinvexity, duality theorems are stated. We give weak duality theorems, proving that the value of the objective function of the primal cannot exceed the value of the dual. Moreover, we study the connection between values of the objective functions of the primal and dual programs, in direct and converse duality theorems. While the results in §1 and §2 are introductory in nature, to the best of our knowledge, the results in §3 are new and they have not been reported in literature.

Meso-scale oriented simulation towards virtual process engineering (VPE)—The EMMS Paradigm

With the dramatic development of computational science and technology, computer simulation is playing an increasingly important role in scientific research and engineering practice, and is believed to bring about a profound revolution to the mode and means of these activities. For chemical engineering, it will promote the transition from an experience-and-experiment-based research and development mode to the one based on virtual process engineering (VPE). However, such a revolution still requires tremendous improvements in the Accuracy of physical modeling and numerical methods, the Capability of the computing hardware and software, and the Efficiency of the simulation activities, or in short, ACE. This article will systematically review the 3-decade endeavors at IPE, CAS (Institute of Process Engineering, Chinese Academy of Sciences) on upgrading ACE by establishing a multi-scale computing paradigm focusing on meso-scale structures. We also report recent developments in this direction with projections on future work. Meso-scales refer to the intermediate scales at which the discrete elements in a system interact to shape the global behavior of the system. Reasonable description of the structures at such scales is a bottleneck for reliable and accurate modeling of global behaviors. To address this problem, we started from gas–solid flow by proposing the energy minimization multi-scale (EMMS) model to quantify the hydrodynamics of meso-scale structures in gas–solid fluidization. This model, that is, its stability condition, was thoroughly analyzed and verified with the pseudo-particle modeling (PPM) method. Other systems, such as turbulence and gas–liquid flow, were then studied following the same strategy, revealing the ubiquity of stability conditions for meso-scale structures as a result of the compromise among different dominant mechanisms, and leading to a generalized model mathematically formulated as a multi-objective variational (MOV) problem. Meanwhile, along with wider industrial applications of the EMMS-based simulation methods, common features were recognized that ACE requires structural consistency among the simulated system, the physical model, the numerical software and the computing hardware under the umbrella of MOV, which we have named as the EMMS Paradigm. The EMMS Paradigm was first implemented from the software level using traditional computing hardware to achieve Accuracy. Since 2007, with the development of general-purpose GPU (graphic processing unit) computing, CPU (central processing unit)–GPU hybrid computing was deployed to implement the Paradigm from the hardware level, boosting the Capability to Petaflops range. In applying the Paradigm to the development of an industrial petro-chemical process (maximizing iso-paraffins, MIP), enabling technologies such as pre- and post-processing, error-tolerance, network-based computing and seamless integration of different simulation methods are now under development, which will greatly improve the Efficiency. A demonstration virtual laboratory featuring the comparison and interaction of online measurement and real-time computing is under construction. With these endeavors, an industrial process could be simulated globally in almost real-time, and different levels of details could be traced from the global distribution of flow parameters in a reactor to the inner-channels of catalytic particles at a perceivable evolution speed. The realization of VPE is, therefore, in the foreseeable future.

Weak efficiency in multiobjective variational problems under generalized convexity

In this paper, we provide new pseudoinvexity conditions on the involved functionals of a multiobjective variational problem, such that all vector Kuhn-Tucker or Fritz John points are weakly efficient solutions if and only if these conditions are fulfilled. We relate weakly efficient solutions to optimal solutions of weighting problems. We improve recent papers, and we generalize pseudoinvexity conditions used in multiobjective mathematical programming, so as some of their characterization results. The new conditions and results are illustrated with an example.

Minimization of vectors of curvilinear functionals on the second order jet bundle

Consider the multi-time multi-objective variational problem (MFP) of minimizing a vector of quotients of path independent curvilinear functionals subject to PDE and/or PDI constraints. The goal of our work is to develop an optimization theory for the second order jet bundle. While the background in Sect. 1 is introductory, the theory in Sects. 2 and 3 is new as a whole, containing our results.

Optimality Conditions and Duality for Multiobjective Variational Problems with Generalized ρ − (η, θ)‐B‐Type‐I Functions

We use ρ − (η, θ)‐B‐type‐I and generalized ρ − (η, θ)‐B‐type‐I functions to establish sufficient optimality conditions and duality results for multiobjective variational problems. Some of the related problems are also discussed.

A necessary and sufficient condition for duality in multiobjective variational problems

Abstract In this paper we move forward in the study of duality and efficiency in multiobjective variational problems. We introduce new classes of pseudoinvex functions, and prove that not only it is a sufficient condition to establish duality results, but it is also necessary. Moreover, these functions are characterized in order that all Kuhn–Tucker or Fritz John points are efficient solutions. Recent papers are improved. We provide an example to show this improvement and illustrate these classes of functions and results.

Multiobjective Duality in Variational Problems with Higher Order Derivatives

A multiobjective variational problem involving higher order derivatives is considered and optimality condi-tions for this problem are derived. A Mond-Weir type dual to this problem is constructed and various duality results are validated under generalized invexity. Some special cases are mentioned and it is also pointed out that our results can be considered as a dynamic generalization of the already existing results in nonlinear programming.

Sufficiency and duality for multiobjective variational control problems with generalized [formula omitted]-V-convexity

Zalmai introduced generalized ( F , α , ρ , θ ) -V-convex functions, a new class of functions that unifies several concepts of generalized convex functions. In this article, the generalized ( F , α , ρ , θ ) -V-convex functions are extended to variational control problem. By utilizing the new concepts, we obtain sufficient optimality conditions and prove Wolfe type and Mond–Weir type duality results for multiobjective variational control programming problem.

Symmetric duality for minimax mixed integer programming problems with pseudo-invexity

In this paper, we consider the pair of symmetric dual multiobjective variational mixed integer programs proposed by Chen and Yang [X. Chen, J. Yang, Symmetric duality for minimax multiobjective variational mixed integer programming problems with partial-invexity, European Journal of Operational Research 181 (2007) 76-85.] and extend some of their results under the assumptions of partial-pseudo-invexity and separability on the functions involved. These results include several results available in literature as special cases.

Necessary and sufficient conditions for local Pareto optimality on time scales

We study a multiobjective variational problem on time scales. For this problem, necessary and sufficient conditions for weak local Pareto optimality are given. We also prove a necessary optimality condition for the isoperimetric problem with multiple constraints on time scales.

Multiobjective Variational Programming under Generalized Type I Univexity

In this paper, we extend the Mishra, Rueda and Giorgi generalized V-univexity type I, defined for multiobjective programming, to multiobjective variational programming problems and we derive various sufficient optimality conditions and mixed type duality results under generalized V-univexity type I conditions.

Symmetric duality for multiobjective fractional variational problems involving cones

In this paper, a pair of multiobjective fractional variational symmetric dual problems over cones is formulated. Weak, strong and converse duality theorems are established under generalized F-convexity assumptions. Moreover, self duality theorem is also discussed.

Symmetric duality for minimax multiobjective variational mixed integer programming problems with partial-invexity

A pair of symmetric dual multiobjective variational mixed integer programs for the polars of arbitrary cones are formulated, which some primal and dual variables are constrained to belong to the set of integers. Under the separability with respect to integer variables and partial-invexity assumptions on the functions involved, we prove the weak, strong, converse and self-duality theorems to related minimax efficient solution. These results include some of available results.

Duality for multiobjective variational control and multiobjective fractional variational control problems with pseudoinvexity

A class of multiobjective variational control and multiobjective fractional variational control problems is considered, and the duality results are formulated. Under pseudoinvexity assumptions on the functions involved, weak, strong, and converse duality theorems are proved.

Sufficient optimality criteria and duality for multiobjective variational control problems with [formula omitted]-invexity

The sufficient optimality criteria for a class of multiobjective variational control problems under the V -invexity assumption are presented. Duality results are proved under a variety of V -invexity assumptions. Multiobjective fractional variational control problems are also studied.

Sufficiency and Duality in Multiobjective Variational Problems with Generalized Type I Functions

Recently Hachimi and Aghezzaf introduced the notion of (F,α,ρ,d)-type I functions, a new class of functions that unifies several concepts of generalized type I functions. Here, we extend the concepts of (F,α,ρ,d)-type I and generalized (F,α,ρ,d)-type I functions to the continuous case and we use these concepts to establish various sufficient optimality conditions and mixed duality results for multiobjective variational problems. Our results apparently generalize a fairly large number of sufficient optimality conditions and duality results previously obtained for multiobjective variational problems.