Multiobjective Variational Problems Research Articles

$ E $-Optimality and $ E $-duality results for multiobjective variational problems and application to the cake-eating problem

$ E $-Optimality and $ E $-duality results for multiobjective variational problems and application to the cake-eating problem

On a minimal criterion of efficiency in vector variational control problems

In this paper, we establish a minimal criterion of efficiency for a class of multiobjective variational control problems. Thus, some minimal conditions are formulated so that all local efficient solutions are also global. Also, we present an example to illustrate the mathematical developments derived in the paper.

Exploration on the stability conditions in bubble columns by noncooperative game theory

• Different pathways of compromise of dominant mechanisms in bubbly flow. • Formulation of different stability conditions for gas–iquid bubbly flow. • Game theory and non-inferior solution to unravel the compromising mechanisms. • Understand regime transition and structure of bubble classes via stability condition. The energy-minimization multiscale (EMMS) model, originally proposed for gas–solid fluidization, features a stability condition to close the simplified conservation equations. It was put forward to physically reflect the compromise of two dominant mechanisms, i.e. , the particle-dominated with minimal potential energy of particles, and the gas-dominated with the least resistance for gas to penetrate through the particle bed. The stability condition was then formulated as the minimization of the ratio of these two physical quantities. Analogously, the EMMS approach was later extended to the gas–liquid flow in bubble columns, termed dual-bubble-size model. It considers the compromise of two dominant mechanisms, i.e ., the liquid-dominated regime with small bubbles, and the gas-dominated regime with large bubbles. The stability condition was then formulated as the minimization of the sum of these two physical quantities. Obviously, the two stability conditions were expressed in different manner, though gas–solid and gas–liquid systems bear some analogy. In addition, both the conditions transform the original multi-objective variational problem into a single-objective problem. The mathematical formulation of stability condition remains therefore an open question. This study utilizes noncooperative game theory and noninferior solutions to directly solve the multi-objective variational problem, aiming to explore the different pathways of compromise of dominant mechanisms. The results show that only keeping the single dominant mechanism cannot capture the jump change of gas holdup, which is associated with flow regime transition. Hybrid of dominant mechanisms, noninferior solutions and noncooperative game theory can predict the flow regime transition. However, the game between the two mechanisms makes the two-bubble structure degenerate and reduce to the single-bubble structure. The game of the three mechanisms restores the two-bubble structure. The exploration on the formulation of stability conditions may help to understand the roles and interactions of different dominant mechanisms in the origin of complexity in multiphase flow systems.

Generalized invexity and duality in multiobjective variational problems involving non-singular fractional derivative

AbstractIn this article, we extend the generalized invexity and duality results for multiobjective variational problems with fractional derivative pertaining to an exponential kernel by using the concept of weak minima. Multiobjective variational problems find their applications in economic planning, flight control design, industrial process control, control of space structures, control of production and inventory, advertising investment, impulsive control problems, mechanics, and several other engineering and scientific problems. The proposed work considers the newly derived Caputo–Fabrizio (CF) fractional derivative operator. It is actually a convolution of the exponential function and the first-order derivative. The significant characteristic of this fractional derivative operator is that it provides a non-singular exponential kernel, which describes the dynamics of a system in a better way. Moreover, the proposed work also presents various weak, strong, and converse duality theorems under the diverse generalized invexity conditions in view of the CF fractional derivative operator.

Parametric approach for a class of fractional variational programs involving support functions

In this paper, a pair of non-differentiable second-order multi-objective symmetrical dual fractional variational programs over arbitrary cones is constructed. We have introduced the concept of second-order (F, α, ρ, d)-pseudoconvexity and an example of a function which is second-order (F, α, ρ, d)-pseudoconvex but not second-order F-pseudoconvex has been constructed. Further, to relate the objective function value of primal and dual, we proved weak, strong, and converse duality theorems under second-order (F, α, ρ, d)-pseudoconvexity suppositions. Moreover, to validate the results of the weak duality theorem, an example of a multi-objective variational fractional problem and its dual model has been constructed. Several known results are also discussed as special cases.

Multitime multiobjective variational problems via η-approximation method

The present article is devoted to multitime multiobjective variational problems via ?-approximation method. In this method, an ?-approximation approach is applied to the considered problem, and a new problem is constructed, called as ?- approximated multitime multiobjective variational problem that contains the change in objective and both constraints functions. The equivalence between an efficient (Pareto optimal) solution to the main multitime multiobjective variational problem is derived along with its associated ?-approximated problem under invexity defined for a multitime functional. Furthermore, we have also discussed the saddle-point criteria for the problem considered and its associated ?-approximated problems via generalized invexity assumptions.

Characterization of [formula omitted]-efficiency and saddle-point criteria for [formula omitted]-approximated multiobjective interval-valued variational problems

This paper aims to find a solution of interval uncertainty to multiobjective variational problems. For this, we consider an interval-valued multiobjective variational problem. Then, by using the modified F-objective function method, we construct associated interval-valued multiobjective variational problem with the modified F-objective functions. We establish a relationship between LU-pareto optimal solution of original problem and its associated modified problem by using the concept of LU-F-convexity and LU-F-pseudoconvexity. Further, we define LU-Lagrange function and its saddle point to discuss the efficient solution of original problem through it. We provide an example to validate our results numerically.

On a Dual Pair of Multiobjective Interval-Valued Variational Control Problems

In this paper, by using the new concept of (ϱ,ψ,ω)-quasiinvexity associated with interval-valued path-independent curvilinear integral functionals, we establish some duality results for a new class of multiobjective variational control problems with interval-valued components. More concretely, we formulate and prove weak, strong, and converse duality theorems under (ϱ,ψ,ω)-quasiinvexity hypotheses for the considered class of optimization problems.

Duality for multiobjective variational problems under second-order (Ф,ρ)-invexity

The purpose of this article is to introduce the concept of second order (?,?)-invex function for continuous case and apply it to discuss the duality relations for a class of multiobjective variational problem. Weak, strong and strict duality theorems are obtained in order to relate efficient solutions of the primal problem and its second order Mond-Weir type multiobjective variational dual problem using aforesaid assumption. A non-trivial example is also exemplified to show the presence of the proposed class of a function.

On a global efficiency criterion in multiobjective variational control problems with path-independent curvilinear integral cost functionals

In this paper, a global efficiency criterion is established for a class of multidimensional variational control problems governed by first order PDE&PDI constraints and path-independent curvilinear integral cost functionals. More precisely, a minimal efficiency criterion for a local efficient solution to be its global efficient solution in the considered optimization problem is formulated and proved. Also, the theoretical developments derived in the paper are accompanied by an example of a nonconvex optimization problem.

Second-order symmetric duality in multiobjective variational problems

In this work, we introduce a pair of multiobjective second-order symmetric dual variational problems. Weak, strong, and converse duality theorems for this pair are established under the assumption of ?-bonvexity/?-pseudobonvexity. At the end, the static case of our problems has also been discussed.

Second order duality for variational problem via efficiency of higher order

A mixed type second order dual of a multiobjective variational problem has been considered. Notion of generalised second order (F, ρ, θ, m)-invexity is introduced which is utilised to obtain duality results using efficiency of higher order as optimality criteria. Generalised second order (F, ρ, θ, m)-invexity assumptions broadens the domain of the problem, whereas efficiency of higher order leads to stronger results.

Multiobjective variational problems with time delay

In this paper, the study of multiobjective variational problems with time delay is introduced. We provide methods for identifying Pareto optimal solutions. We also prove necessary and sufficient optimality conditions for the isoperimetric problem with multiple constraints and delayed arguments.

On a New Class of Vector Variational Control Problems

In this paper, we introduce geodesic (strongly) b-V-KT-pseudoinvex multidimensional control problems. This new class of multiobjective variational control problems, involving multiple integral cost functionals, is described such that every geodesic Kuhn–Tucker point is geodesic efficient solution. In addition, to illustrate the effectiveness of our main result, the paper is completed with an application.

The Modified Objective Function Method for Univex Multiobjective Variational Problems

In this paper, we use the modified objective function method for a class of nonconvex multiobjective variational problems involving univex functions. Under univexity hypotheses, we prove the equivalence between an (weakly) efficient solution of the considered multiobjective variational problem and an (weakly) efficient solution of the associated modified multiobjective variational problem constructed in the modified objective function method.

Application of the Fast Marching Method for Path Planning of Long-haul Optical Fiber Cables With Shielding

This paper provides a method for optimal shielding design and path planning of a long-haul optical fiber cable between two locations on the Earth’s surface. The method allows minimization of the cable laying cost including material and labor and the risks of future cable break associated with laying the cable through various areas, including earthquake-prone or other risky areas. Both costs per unit length and risk of cable damage may be different at different locations. Expensive shielding may be important in certain high-risk areas and unnecessary in lower risk areas. We use ground motion intensity to estimate future cable repair rate (our measure of earthquake-related cable damage risk), and a triangulated manifold to represent the surface of the Earth. With laying cost and expected total number of repairs of the cable as the two objectives, we formulate the problem as a multiobjective variational optimization problem. This formulation incorporating multiple design levels for cable shielding is converted into a single objective variational optimization problem by assigning different weights to each objective. The solution path of the later problem is obtained by using the Fast Marching Method (FMM) with an additional minimization step. A new proof of the optimality of FMM for the problem is provided. Numerical results demonstrate that the FMM-based method outperforms existing raster-based algorithms. With billions of US dollars spent yearly on new cables, the potential savings are substantial. Furthermore, the computational complexity of the FMM-based method is $O(N \log (N))$ , making it applicable to cables of realistic length.

Multiobjective Path Optimization for Critical Infrastructure Links with Consideration to Seismic Resilience

AbstractWe study the generic problem of path optimization for a critical infrastructure link between two locations on the surface of the Earth in the vicinity of earthquake‐prone areas. The problem has two (conflicting) objective functions, one for minimizing the construction cost of the link and the other for minimizing the number of potential repairs along the link in the wake of earthquakes. In our model, the Earth's surface is approximated by a triangulated manifold, and ground motion intensity data are used to provide a measure of repair rate. We approach the multiobjective variational problem by first converting it into a single objective variational problem using the weighted sum method. Then, we show that the problem can be further transformed into an Eikonal equation and solved by a computationally efficient algorithm based on the fast marching method. Extensive simulations are performed on real‐world three‐dimensional geographical data, from which we obtain Pareto optimal solutions that provide insight and guidance to design trade‐offs between cost effectiveness and seismic resilience.

η-Approximation Method for Non-convex Multiobjective Variational Problems

ABSTRACTIn this paper, we use the η-approximation method for a class of non-convex multiobjective variational problems with invex functionals. In this approach, for the considered multiobjective variational problem, the associated η-approximated multiobjective variational problem is constructed at the given feasible solution. The equivalence between (weakly) efficient solutions in the original multiobjective variational problem and its associated η-approximated multiobjective variational problem is established under invexity hypotheses.

Multi-objective semi-infinite variational problem and generalized invexity

In this paper, multiobjective semi-infinite variational problem (MSVP) has been considered. Relationship between efficiency, vector Kuhn Tucker point and vector Fritz John point for the stated problem have been established and authenticated by means of examples. Two duals namely Wolfe and Mond–Weir are purposed for (MSVP) and duality results are established under generalized invexity assumptions. Efficient solution of (MSVP) has been characterized by vector saddle point of a vector valued Lagrangian function of (MSVP). An equivalent (MSVP) is also constructed by a suitable modification of the objective function.

Multiobjective variational problems and generalized vector variational-type inequalities

The purpose of this paper is to generalize the vector variational-type inequalities, formulated by Kim [J. Appl. Math. Comput. 16 (2004) 279–287], by setting the norms into Minty and Stampacchia forms. We also demonstrate the relationships between these generalized inequalities and multiobjective variational problems, by using the notions of strongly convex functionals. The theoretical developments are illustrated through numerical examples.