Stochastic Gradient Descent (SGD) is the cornerstone of large-scale optimization; however, its application to problems with a vast number of constraints remains a significant challenge. Methods based on relaxed logarithmic barrier functions have enabled the use of SGD by sampling constraints, yet the reliance on single samples in these methods often leads to high variance and oscillations that hinder progress near the optimal solution.In this paper, we propose a novel hybrid algorithm, AdaBS-RBSGD, which integrates the relaxed barrier framework with an adaptive batch size strategy. Our algorithm dynamically adjusts the batch size Bk to be inversely proportional to the estimated optimization loss, systematically reducing gradient variance as the iterations approach the solution.We provide a rigorous mathematical proof establishing the almost-sure convergence of this hybrid algorithm. Furthermore, our theoretical analysis of error bounds demonstrates that active variance management allows the error term associated with stochastic noise to vanish more rapidly compared to the baseline method. Empirically, results on a benchmark problem show that the proposed algorithm significantly outperforms its single-sample counterpart, successfully breaking the “noise floor,” achieving higher stability, and reaching a more accurate final solution. This work presents a practical and theoretically sound methodology for accelerating and improving the precision of stochastic constrained optimization.
In this article, we study a singular fractional-order Caputo differential problem involving boundary conditions with linear operators. The analysis is carried out in an appropriate Banach space framework. First, by applying the Banach contraction principle, we establish a uniqueness result for the considered problem. Then, using Krasnosel'skii's fixed point theorem, we derive an existence theorem under suitable assumptions on the nonlinear term. In the second part of the paper, we investigate different types of stability for the obtained solutions, namely Ulam-Hyers stability and Ulam-Hyers-Rassias stability. Sufficient conditions ensuring these stability properties are presented and proved. The obtained theoretical results highlight the influence of the singular fractional operator and the associated boundary conditions. Finally, two illustrative examples are discussed in order to demonstrate the applicability and effectiveness of the established results.
This paper proposes a novel common-weight DEA model that combines multi-objective linear fractional programming and goal programming to determine a set of common-weights for evaluating decision-making units (DMUs). The model formulates a single linear program (LP) without requiring iterative procedures, thereby enhancing computational efficiency. To validate the model’s effectiveness, we compare the proposed method with a satisfaction degree–based model, a distance-based model, and a credibility-based common-weight method. Our model demonstrates competitive efficiency scores and a high degree of discrimination among DMUs, while maintaining a lower computational time complexity. This research contributes by offering a practical and computationally efficient alternative to existing models.
We study an inverse eigenvalue problem associated with a special class of structured matrices in both symmetric and nonsymmetric forms. In the symmetric case, the problem involves two sets of eigenpairs corresponding to the original matrix and its leading principal submatrix of order $n-1$. For the nonsymmetric case, the minimum and maximum eigenvalues of each leading principal submatrix, together with an eigenpair associated with the maximum eigenvalue of the original matrix, are prescribed. By establishing recurrence relations between successive principal submatrices, we derive the necessary and sufficient conditions for the solvability of the problems. Moreover, explicit reconstruction algorithms are developed, and several numerical results are presented to illustrate the effectiveness of the proposed methods.
This paper examines the applicability of a new exponentially fitted scheme on Shishkin mesh for solving a class of singularly perturbed problems in ordinary differential equations containing a small delayed term. The original delayed problem is reduced to an equivalent new problem using the Taylor series expansion procedure, and then a three-term recurrence relationship is obtained by approximating the first and second order derivatives in the reduced problem by their mixed and central difference analogue, respectively. To avoid the non-uniformity in the solution, a fitting factor is introduced into the obtained tri-diagonal scheme, and its value is determined by the use of singular perturbation theory. Thomas' algorithm is efficiently used to solve tri-diagonal system of equations. Convergence and stability of the scheme are discussed in detail. Richardson's extrapolation technique is employed to improve the accuracy and rate of convergence of the solution produced by the scheme. Four standard example problems are solved, and the computational results are presented in terms of the maximum absolute point-wise errors and rate of convergence. Computational results are compared with some other published results. Theoretical and computational analysis reveal that the fitted scheme is able to produce accurate and uniformly convergent solutions with first-order accuracy, which is further improved to second-order accuracy by Richardson extrapolation. Further, the method is most suitable for all the values of (1/N) ≫ ε where ε ≤ 10−6 and N is the number of sub-intervals in which the underlying interval is divided.
