One of the peculiar features of multi-parametric approach for bilevel programs is that most methods using this approach can be extended to tri-level (and generally to $k$-level) programs, which is not always the case with other non-heuristic solution methods. However, most of existing multi-parametric methods work well when the constraint of the lower-level problem is polyhedral. In this article we propose a multi-parametric programming based solution algorithm for bilevel optimization problems whose lower-level problem involves convex smooth nonlinear constraints. The method is also extended to solve some classes of $k$-level convex optimization problems with nonlinear constraints. The algorithm recasts the lower-level problem as a multi-parametric problem and employs an equivalent barrier problem reformulation. The solution obtained through multi-parametric programming is incorporated in the upper-level problem to create a set of single-level optimization problems which are solved using standard global optimization techniques. The proposed algorithm can give an exact global solution to some class of nonlinear Multi-level problems with convex nonlinear constraints.
This work proposes a class of Multi-Objective Moreau Envelope based Hessian-free Algorithms (MOMEHA) to solve the multi-objective bilevel learning problems with nonconvex lower level and proposes a momentum-based variant of MOMEHA (i.e., MB-MOMEHA) method to solve the stochastic multi-objective bilevel learning problems.
For solving nonconvex equality-constrained optimization problems, a recent Gradient-Eigenstep Algorithm by Goyens et al.~is an iteration-efficient approach, based on minimizing Fletcher's augmented Lagrangian function, for finding an approximate second-order stationary point from an arbitrary starting point. In this paper, the analysis of this algorithm is extended, offering a two-fold contribution. First, it is shown that a local-linear rate of convergence can be obtained by this method if it is initiated sufficiently close to a strong second-order stationary point and employs a sufficiently small step-size parameter and sufficiently large penalty parameter. In this case, the algorithm reduces to a gradient descent algorithm applied to minimize Fletcher's augmented Lagrangian. Second, as a particularly useful application of the first result, it is shown that the Gradient-Eigenstep algorithm can be used as an iteration-efficient subproblem solver in the context of a progressive sampling strategy for solving equality-constrained optimization problems when the objective and constraint functions are defined by large sample averages, ultimately offering an algorithm with an improved worst-case sample complexity when compared to an approach that solves a full-sample problem directly.
Frank E. Curtis, Ling-Jun Guo, Daniel P. Robinson· 0 citations
Phase equilibrium problems are central to chemical engineering, underpinning tasks ranging from separation process design to the development of thermodynamic models. A particularly challenging computational task is the generation of phase envelopes: rigorously fitting thermodynamic models to real world data with well behaved predictions requires solving a computationally expensive bilevel optimization problem. We present a geometric reformulation of this parameter estimation problem that restates the bilevel program as a single level problem that is significantly easier to solve. The solution of the single level problem is proven to be the globally optimal solution of the bilevel problem when specialized global optimization solvers are used. In addition, the method retains the constraints that guarantee a well behaved fitted model, such as enforcing the correct number of phase splits, excluding spurious phases, and ensuring stability in regions of instability. This allows the practitioner to reliably and efficiently fit mathematically complex thermodynamic models to data, and potentially enables highly accurate and rigorous modelling of problems in computational thermodynamics that were previously intractable. Finally, an algorithm is presented that is proven to converge for any black box thermodynamic model. Only an expression of the Gibbs free energy is required, no derivatives are needed, and convergence is guaranteed for the broadest class of non-smooth, non-continuous models.
Structural optimization problems often involve a large number of decision variables and highly non-convex feasible regions, making convergence to the true Pareto front extremely challenging. Even when convergence is achievable, it typically requires thousands of function evaluations, resulting in significant computational cost. This highlights the need for efficient and robust optimization algorithms for real-world engineering applications. In this study, we introduce a novel constrained multi-objective evolutionary algorithm, termed DPCME. The algorithm employs two interacting populations that exchange information, enabling effective global exploration and reducing the risk of convergence to local optima. To further enhance performance, a recent repair-based constraint-handling technique is incorporated, and alternative repair approaches are proposed and systematically evaluated. The proposed algorithm is tested on three engineering problems: the 72-bar truss, the 120-bar truss, and a chemical tanker structure, each involving hundreds of nonlinear failure constraints. Its performance is evaluated against state-of-the-art constrained multi-objective optimization algorithms from the latest PlatEMO package. A total of 43 algorithms are initially tested, from which the 12 best-performing methods are selected for detailed comparison. The results demonstrate that DPCME achieves superior or competitive convergence and diversity across all test cases, and that the inclusion of repair-based constraint handling further improves its performance.