This work introduces an adaptive line-search-free multiobjective gradient (AMG) method for solving smooth multiobjective optimization problems. The proposed approach automatically adjusts stepsizes based on steepest descent directions, promoting robustness with respect to stepsize choice while maintaining low computational cost. The method is specifically tailored to the multiobjective setting and does not rely on function evaluations, making it well-suited for this scenario. The proposed algorithm admits two variants: (i) a conservative variant, in which the stepsize is monotonically decreasing; and (ii) a flexible variant, which allows occasional increases in the stepsize. From a theoretical standpoint, under standard Lipschitz continuity assumptions on the gradients, we establish iteration-complexity bounds for achieving a Pareto critical point for both variants in the nonconvex setting. In the convex setting, we further derive improved iteration-complexity bounds for the conservative AMG variant. From a practical standpoint, the numerical experiments demonstrate that the flexible AMG performs favorably compared to the steepest descent method with either a fixed stepsize or Armijo line search.
This study introduces a proximal three-block Alternating Direction Method of Multipliers (ADMM) variant with a projected dual update for solving two-stage stochastic programming problems. We first recast the two-stage stochastic program into a deterministic three-block separable optimization problem, a formulation designed to leverage the parallel computation capabilities inherent in ADMM. A key contribution is the establishment of theoretical convergence for our proposed regularized ADMM variant when applied to this specific three-block formulation; this analysis utilizes principles of lower semi-continuity and the Kurdyka-Lojasiewicz (K-L) property. The parallel implementation of this approach significantly accelerates the solution process, which is crucial for effectively handling large-scale stochastic programming problems. Through numerical analysis on a suite of benchmark problems, including the large-scale storm instance and a non-convex example, this study validates the robust performance of the proposed ADMM approach. Comparisons with Progressive Hedging, a proximal ADMM variant, and direct Gurobi solution of the extensive form highlight its efficiency in terms of computational time and solution quality, particularly for larger problems. This research, therefore, contributes a theoretically sound and practically effective ADMM framework for two-stage stochastic programming, offering valuable insights for broader optimization contexts.
The discretization is an important approach to solve stochastic optimization. In this paper, the discrete approximation for a class of stochastic hierarchical optimization problems (SHOPs) that cannot be recast as dynamic forms is studied. Due to the failure of the reformulation, the classical sample average approximation (SAA) approach cannot be applied to the SHOP directly. Firstly, by invoking the set-valued SAA method, we give a discrete approximation scheme of the SHOP. We verify that the optimal value and the here-and-now solution of the SAA problem converge to those of the SHOP. However, it fails to give a reliable approximation of the wait-and-see solution, which is sometimes important. Alternatively, by replacing the wait-and-see solution with a simple function and replacing the expectation of a random function with a specific realization of the random function, we obtain another discrete approximation counterpart of the SHOP, which can provide a reliable approximation of the wait-and-see solution. We verify that the optimal value and the policy of the discretization problem converge to those of the SHOP. Finally, some illustrative numerical results are reported to validate our discrete approximation schemes.
In this paper, we address the second-order cone complementarity problems (denoted as SOCCPs). Initially, we construct a new class of smooth functions to reformulate SOCCPs as smooth systems of equations. Based on this smoothing functions, we develop a smoothing Levenberg–Marquardt method equipped with a nonmonotone derivative-free line search mechanism. Furthermore, we establish the well-definedness and global convergence of the proposed method. For analysing its local convergence, we employ a local error bound condition, which is a less stringent requirement than the typical nonsingularity assumptions. Finally, numerical results support the theoretical findings and illustrate the efficiency of the method for solving SOCCPs.
We introduce a golden ratio proximal algorithm for pseudo-monotone equilibrium problems in Hilbert spaces that resolves three fundamental limitations of existing single-proximal methods: the need for strong pseudo-monotonicity to guarantee strong convergence, non-increasing step sizes, and confinement to the classical bound ϕ≤ (1+√(5))/2 . Our method uses only one proximal evaluation per iteration and employs a self-adaptive step-size rule that becomes non-decreasing after finitely many steps. We extend the admissible golden ratio parameter to the sharp interval (1,17+√(1105)/24)≈ (1,2.0934) , proving that for any larger ϕ the non-adaptive constant-step-size scheme fails. We establish the first quantitative best-iterate residual estimates O(log k/k) for this class, prove a matching lower bound showing the logarithmic factor is unavoidable, and derive R-linear convergence under strong pseudo-monotonicity. The framework is further extended to strongly quasiconvex problems in finite dimensions. Comprehensive numerical experiments demonstrate consistent superiority over six existing methods in both iteration count and CPU time.
