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Debdas Ghosh

Publications and source records attributed to Debdas Ghosh.

At least 19 recordsLinked to original sources
arXiv2026

Proximal Gradient Methods for Unconstrained Set Optimization Problems with Set-Valued Maps of Finite Cardinality

This work presents two different types of proximal gradient methods, with line search and without line search, for solving unconstrained set-valued optimization problems under the lower set-less ordering relation induced by a solid cone that is convex, pointed, and closed. The objective mapping of the problem involves finitely many functions, with each one being the sum of a continuously differentiable function and a convex function that is proper and closed. We present an approach to characterize weakly minimal points of the problem with the help of weakly efficient points of a family of vector optimization problems. Thereafter, we establish a stationarity condition along with its connection with weakly minimal points of the problem under study. Based on the stationary condition, the concept of a descent direction at a non-stationary point is discussed. In view of the line search-based method, we formulate an Armijo-type line search condition and establish the existence of such a step-size. For the proposed methods, global convergence is established under mild assumptions. The convergence analysis of the proximal gradient method with line search provides a theoretical advancement over the convergence results previously established for the steepest descent method in set-valued optimization problems. In addition, we analyze the computational complexity of the proposed methods and show that both methods achieve a convergence rate of $\mathcal{O}(1/\sqrt{k})$. Numerical results are reported to test the performance of the methods in practice.

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arXiv2026

Hager-Zhang Conjugate Gradient Method for Set Optimization with Set-Valued Objective Map of Finite Cardinality

This work introduces a nonlinear Hager-Zhang conjugate gradient method for solving set optimization problems. The objective function under consideration is defined by a finite collection of continuously differentiable functions. Notably, the proposed approach imposes restrictions neither on the existence of a finite generator of the ordering cone nor on any regularity condition at the optimal solution. As a result, the proposed method holds considerable significance for both set optimization and vector optimization problems, with the latter serving as a special case of the former. The study begins by discussing Wolfe line search conditions using Drummond-Svaiter scalarization function. Thereafter, we establish the existence of a step length satisfying the Wolfe line search conditions along a descent direction. The Hager-Zhang scalar conjugate parameter is introduced to derive the search direction for the proposed method. It is established that the direction generated by the proposed method is a descent direction. The well-definedness of the proposed method is given. Furthermore, we discuss some important results and a Zoutendijk-like condition to ensure global convergence. Subsequently, the global convergence of the proposed method is established in an asymptotic manner. Finally, numerical experiments on various test problems validate the practical performance and effectiveness of the proposed technique.

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arXiv2026

Bayesian fuzzy optimization using fuzzy Gaussian process

In many real-life problems, decision-making gets complicated due to dual sources of uncertainty, known as randomness and fuzziness or imprecision, which can be challenging for traditional optimization methods. Most of the existing fuzzy optimization techniques that optimize fuzzy-valued objective functions ignore fuzziness, while the probabilistic optimization techniques ignore randomness. To handle this dual source of uncertainty, Kwakernaak introduced the concept of a fuzzy random variable as ``random variables whose values are not real, but fuzzy numbers". This work aims to derive a theoretical background for the Gaussian fuzzy process and fuzzy acquisition functions, which will be used to develop a novel \emph{Bayesian fuzzy optimization} (BFO) technique that optimizes a fuzzy-values objective function. Based on fuzzy random variables, the Gaussian fuzzy process is developed, which is used as a prior belief about the fuzzy-valued objective function in the BFO. Fuzzy acquisition functions are defined to act as a guide for the search process of BFO with the help of posterior fuzzy mean and fuzzy variance. The proposed method demonstrated effective performance in both fuzzy mean-variance portfolio allocation and Indian temperature data analysis, showing robust predictive accuracy and adaptability. The proposed method can have broader applications in various fields like healthcare, material science, agriculture, etc.

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arXiv2026

Nonlinear Conjugate Gradient Method for Multiobjective Optimization Problems of Interval-Valued Maps

In this article, we propose an algorithm for the nonlinear conjugate gradient method to find a Pareto critical point of unconstrained multiobjective interval optimization problems. In this algorithm, we use the Wolfe line search procedure to find the step length. After defining the standard Wolfe conditions and the strong Wolfe conditions, we prove that there exists an interval of the step length that satisfies the standard Wolfe conditions and the strong Wolfe conditions. Further, to study the convergence analysis of our proposed algorithm, we derive the result related to the Zoutendijk condition. In the convergence analysis, first, we prove the global convergence property of our proposed algorithm for a general conjugate gradient algorithmic parameter. Further, we consider four variants of the conjugate gradient algorithmic parameter, such as Fletcher-Reeves, conjugate descent, Dai-Yuan, and modified Dai-Yuan. For each variant of the algorithmic parameter, we prove the global convergence results of our proposed algorithm. Finally, we test our algorithm on some test problems and make a performance profile.

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arXiv2026

Newton Method for Multiobjective Optimization Problems of Interval-Valued Maps

In this article, we propose a Newton-based method for solving multiobjective interval optimization problems (MIOPs). We first provide a connection between weakly Pareto optimal points and Pareto critical points in the context of MIOPs. Introducing this relationship, we develop an algorithm aimed at computing a Pareto critical point. The algorithm incorporates the computation of a descent direction at a non-Pareto critical point and employs an Armijo-like line search strategy to ensure sufficient decrease. Under suitable assumptions, we prove that the sequence generated by our proposed algorithm converges to a Pareto critical point. The effectiveness and performance of the proposed method are demonstrated through a series of numerical experiments on some test problems. Finally, we apply our proposed algorithm in a portfolio optimization problem with interval uncertainty.

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arXiv2025

Nonmonotone Trust-Region Methods for Optimization of Set-Valued Mapping of Finite Cardinality

Non-monotone trust-region methods are known to provide additional benefits for scalar and multi-objective optimization, such as enhancing the probability of convergence and improving the speed of convergence. For optimization of set-valued maps, non-monotone trust-region methods have not yet been explored and investigated to see if they show similar benefits. Thus, in this article, we propose two non-monotone trust-region schemes--max-type and average-type for set-valued optimization. Using these methods, the aim is to find \emph{K}-critical points for a non-convex unconstrained set optimization problem through vectorization and oriented-distance scalarization. The main modification in the existing trust region method for set optimization occurs in reduction ratios, where max-type uses the maximum over function values from the last few iterations, and avg-type uses an exponentially weighted moving average of successive previous function values till the current iteration. Under appropriate assumptions, we show the global convergence of the proposed methods. To verify their effectiveness, we numerically compare their performance with the existing trust region method, steepest descent method, and conjugate gradient method using performance profile in terms of three metrics: number of non-convergence, number of iterations, and computation time.

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arXiv2025

Trust-Region Method for Optimization of Set-Valued Maps Given by Finitely Many Functions

In this article, we develop a trust-region technique to find critical points of unconstrained set optimization problems with the objective set-valued map defined by finitely many twice continuously differentiable functions. The technique is globally convergent and has the descent property. To ensure the descent property, a new rule of trust-region reduction ratio is introduced for the considered set-valued maps. In the derived method, to find the sequence of iteration points, we need to perform one iteration of a different vector optimization problem at each iteration. Thus, the derived technique is found to be not a straight extension of that for vector optimization. The effectiveness of the proposed algorithm is reported through performance profiles of the proposed approach with the existing methods on various test examples. A list of test problems for set optimization is also provided.

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arXiv2025

Cubic Regularization Technique of the Newton Method for Vector Optimization

This study proposes a cubic regularization of the Newton method for generating weakly efficient points of unconstrained vector optimization problems under no convexity assumption on the objective function. It is observed that at a given iterate, the cubic regularized Newton direction is not necessarily a descent direction. In generating the sequence of iterates, no line search is utilized to find a suitable step length to move along the cubic regularized Newton direction. Yet, the proposed method exhibits a global convergence property with $O(k^{-2/3})$ rate of convergence. Further, the local q-quadratic convergence of the Newton method is also retained in the cubic regularization. A new stopping condition is used, which enforces the proposed method to enter in close neighborhood of non-weakly efficient points that are stationary. Thus, the studied technique ends up generating weakly efficient points, not just Pareto critical points. In addition, conditions on the choice of regularization parameter value under which the full cubic regularized Newton step becomes descent are derived. Performance profiles and comparison of the derived method with the existing methods on several test examples are also provided.

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arXiv2025

Global Convergence and Rate Analysis of the Steepest Descent Method for Uncertain Multiobjective Optimization via a Robust Optimization Approach

In this article, we extend our previous work (Applicable Analysis, 2024, pp. 1-25) on the steepest descent method for uncertain multiobjective optimization problems. While that study established local convergence, it did not address global convergence and the rate of convergence of the steepest descent algorithm. To bridge this gap, we provide rigorous proofs for both global convergence and the linear convergence rate of the steepest descent algorithm. Global convergence analysis strengthens the theoretical foundation of the steepest descent method for uncertain multiobjective optimization problems, offering deeper insights into its efficiency and robustness across a broader class of optimization problems. These findings enhance the method's practical applicability and contribute to the advancement of robust optimization techniques.

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arXiv2025

Robust Optimization Approach for Solving Uncertain Multiobjective Optimization Problems Using the Projected Gradient Method

Numerous real-world applications of uncertain multiobjective optimization problems (UMOPs) can be found in science, engineering, business, and management. To handle the solution of uncertain optimization problems, robust optimization is a relatively new field. An extended version of the projected gradient method (PGM) for a deterministic smooth multiobjective optimization problem (MOP) is presented in the current study as a PGM for UMOP. An objective-wise worst-case cost (OWWC) type robust counterpart is considered, and the PGM is used to solve a UMOP by using OWWC. A projected gradient descent algorithm is created using theoretical findings. It is demonstrated that the projected gradient descent algorithm's generated sequence converges to the robust counterpart's weak Pareto optimal solution, which will be the robust weak Pareto optimal solution for UMOP. Under a few reasonable presumptions, the projected gradient descent algorithm's full convergent behavior is also justified. Finally, numerical tests are presented to validate the proposed method.

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arXiv2025

Solution of Uncertain Multiobjective Optimization Problems by Using Nonlinear Conjugate Gradient Method

This paper introduces a nonlinear conjugate gradient method (NCGM) for addressing the robust counterpart of uncertain multiobjective optimization problems (UMOPs). Here, the robust counterpart is defined as the minimum across objective-wise worst-case scenarios. There are some drawbacks to using scalarization techniques to solve the robust counterparts of UMOPs, such as the pre-specification and restrictions of weights, and function importance that is unknown beforehand. NCGM is free from any kind of priori chosen scalars or ordering information of objective functions as accepted in scalarization methods. With the help of NCGM, we determine the critical point for the robust counterpart of UMOP, which is the robust critical point for UMOP. To tackle this robust counterpart using the NCGM, the approach involves constructing and solving a subproblem to determine a descent direction. Subsequently, a new direction is derived based on parameter selection methods such as Fletcher-Reeves, conjugate descent, Dai-Yuan, Polak-Ribi$\grave{e}$re-Polyak, and Hestenes-Stiefel. An Armijo-type inexact line search is employed to identify an appropriate step length. Utilizing descent direction and step length, a sequence is generated, and convergence of the proposed method is established. The effectiveness of the proposed method is verified and compared against an existing method using a set of test problems.

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arXiv2024

Quasi-Newton Method for Set Optimization Problems with Set-Valued Mapping Given by Finitely Many Vector-Valued Functions

In this article, we propose a quasi-Newton method for unconstrained set optimization problems to find its weakly minimal solutions with respect to lower set-less ordering. The set-valued objective mapping under consideration is given by a finite number of vector-valued functions that are twice continuously differentiable. To find the necessary optimality condition for weak minimal points with the help of the proposed quasi-Newton method, we use the concept of partition and formulate a family of vector optimization problems. The evaluation of necessary optimality condition for finding the weakly minimal points involves the computation of the approximate Hessian of every objective function, which is done by a quasi-Newton scheme for vector optimization problems. In the proposed quasi-Newton method, we derive a sequence of iterative points that exhibits convergence to a point which satisfies the derived necessary optimality condition for weakly minimal points. After that, we find a descent direction for a suitably chosen vector optimization problem from this family of vector optimization problems and update from the current iterate to the next iterate. The proposed quasi-Newton method for set optimization problems is not a direct extension of that for vector optimization problems, as the selected vector optimization problem varies across the iterates. The well-definedness and convergence of the proposed method are analyzed. The convergence of the proposed algorithm under some regularity condition of the stationary points, a condition on nonstationary points, the boundedness of the norm of quasi-Newton direction, and the existence of step length that satisfies the Armijo condition are derived. We obtain a local superlinear convergence of the proposed method under uniform continuity of the Hessian approximation function.

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arXiv2024

Nonlinear Conjugate Gradient Methods for Optimization of Set-Valued Mappings of Finite Cardinality

This article presents nonlinear conjugate gradient methods for finding local weakly minimal points of set-valued optimization problems under a lower set less ordering relation. The set-valued objective function of the optimization problem under consideration is defined by finitely many continuously differentiable vector-valued functions. For such optimization problems, at first, we propose a general scheme for nonlinear conjugate gradient methods and then introduce Dai-Yuan, Polak-Ribi{\`e}re-Polyak, and Hestenes-Stiefel conjugate gradient parameters for set-valued functions. Toward deriving the general scheme, we introduce a condition of sufficient decrease and Wolfe line searches for set-valued functions. For a given sequence of descent directions of a set-valued function, it is found that if the proposed standard Wolfe line search technique is employed, then the generated sequence of iterates for set optimization follows a Zoutendijk-like condition. With the help of the derived Zoutendijk-like condition, we report that all the proposed nonlinear conjugate gradient schemes are globally convergent under usual assumptions. It is important to note that the ordering cone used in the entire study is not restricted to be finitely generated, and no regularity assumption on the solution set of the problem is required for any of the reported convergence analyses. Finally, we demonstrate the performance of the proposed methods through numerical experiments. In the numerical experiments, we demonstrate the effectiveness of the proposed methods not only on the commonly used test instances for set optimization but also on a few newly introduced problems under general ordering cones that are neither nonnegative hyper-octant nor finitely generated.

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arXiv2024

A Three-Operator Splitting Scheme Derived from Three-Block ADMM

This work presents a new three-operator splitting method to handle monotone inclusion and convex optimization problems. The proposed splitting serves as another natural extension of the Douglas-Rachford splitting technique to problems involving three operators. For solving a composite convex minimization of a sum of three functions, its formula resembles but is different from Davis-Yin splitting and the dual formulation of the classical three-block ADMM. Numerical tests suggest that such a splitting scheme is robust in the sense of allowing larger step sizes. When two functions have orthogonal domains, the splitting operator can be proven 1/2-averaged, which implies convergence of the iteration scheme using any positive step size.

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arXiv2024

Generalized Ordered Weighted Aggregation Robustness to Solve Uncertain Single Objective Optimization Problems

Robust optimization aims to find optimum points from the collection of points that are feasible for every possible scenario of a given uncertain set. An optimum solution to a robust optimization problem is commonly found by the min-max robust counterpart or by the best out of the worst-cases analysis. In this article, we introduce a new counterpart with the help of the generalized ordered weighted aggregation (GOWA) operator to solve uncertain single objective optimization problems. After introducing GOWA robustness, we analyze a few elementary properties of the GOWA robust objective function, like continuity, monotonicity, coerciveness, local Lipschitz property, and subdifferential regularity. An approach to computing the Clarke subdifferential of the GOWA robust objective function is also provided. We discuss the relationship between the concept of GOWA robustness with other existing robustness -- flimsily, highly, min-max, light, and min-min robustness. We show that in a particular case, GOWA robustness reduces to the commonly used min-max robustness. The entire paper is supported by several geometrical and numerical illustrations.

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arXiv2024

Newton Method for Set Optimization Problems with Set-Valued Mapping of Finitely Many Vector-Valued Functions

In this paper, we propose a Newton method for unconstrained set optimization problems to find its weakly minimal solutions with respect to lower set-less ordering. The objective function of the problem under consideration is given by finitely many strongly convex twice continuously differentiable vector-valued functions. At first, with the help of a family of vector optimization problems and the Gerstewitz scalarizing function, we identify a necessary optimality condition for weakly minimal solutions of the considered problem. In the proposed Newton method, we derive a sequence of iterative points that exhibits local convergence to a point which satisfies the derived necessary optimality condition for weakly minimal points. To find this sequence of iterates, we formulate a family of vector optimization problems with the help of a partition set concept. Then, we find a descent direction for this obtained family of vector optimization problems to progress from the current iterate to the next iterate. As the chosen vector optimization problem differed across the iterates, the proposed Newton method for set optimization problems is not a straight extension of that for vector optimization problems. A step-wise algorithm of the entire process is provided. The well-definedness and convergence of the proposed method are analyzed. To establish the convergence of the proposed algorithm under some regularity condition of the stationary points, we derive three key relations: a condition of nonstationarity, the boundedness of the norm of Newton direction, and the existence of step length that satisfies the Armijo condition. We obtain the local superlinear convergence of the proposed method under uniform continuity of the Hessian and local quadratic convergence under Lipschitz continuity of the Hessian.

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arXiv2024

Second-Order Strong Optimality and Second-Order Duality for Nonsmooth Constrained Multiobjective Fractional Programming Problems

This paper investigates constrained nonsmooth multiobjective fractional programming problem (NMFP) in real Banach spaces. It derives a quotient calculus rule for computing the first- and second-order Clarke derivatives of fractional functions involving locally Lipschitz functions. A novel second-order Abadie-type regularity condition is presented, defined with the help of the Clarke directional derivative and the P\'ales-Zeidan second-order directional derivative. We establish both first- and second-order strong necessary optimality conditions, which contain some new information on multipliers and imply the strong KKT necessary conditions, for a Borwein-type properly efficient solution of NMFP by utilizing generalized directional derivatives. Moreover, it derives second-order sufficient optimality conditions for NMFP under a second-order generalized convexity assumption. Additionally, we derive duality results between NMFP and its second-order dual problem under some appropriate conditions.

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arXiv2023

A Quasi Newton Method for Uncertain Multiobjective Optimization Problems via Robust Optimization Approach

In this paper, we propose a quasi Newton method to solve the robust counterpart of an uncertain multiobjective optimization problem under an arbitrary finite uncertainty set. Here the robust counterpart of an uncertain multiobjective optimization problem is the minimum of objective-wise worst case, which is a nonsmooth deterministic multiobjective optimization problem. In order to solve this robust counterpart with the help of quasi Newton method, we construct a sub-problem using Hessian approximation and solve it to determine a descent direction for the robust counterpart. We introduce an Armijo-type inexact line search technique to find an appropriate step length, and develop a modified BFGS formula to ensure positive definiteness of the Hessian matrix at each iteration. By incorporating descent direction, step length size, and modified BFGS formula, we write the quasi Newton's descent algorithm for the robust counterpart. We prove the convergence of the algorithm under standard assumptions and demonstrate that it achieves superlinear convergence rate. Furthermore, we validate the algorithm by comparing it with the weighted sum method through some numerical examples by using a performance profile.

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