On the Complexity of Subgradient Methods for Trilevel Hierarchical Generalized Variational Inequalities

Categories Convex Optimization, Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

We study generalized variational inequalities with a three-level hierarchical structure. This setting extends nested GVI models beyond the bilevel case, for which $\mathcal{O}(\delta^{-4})$ complexity bounds are known for any prescribed positive tolerance $\delta$, to a fully three-level hierarchical structure. We analyze a projected averaged subgradient method combined with a Tikhonov-like regularization scheme. Under compactness, maximal … Read more

On Stationary Conditions and the Convergence of Augmented Lagrangian methods for Generalized Nash Equilibrium Problems

Categories Constrained Nonlinear Optimization, Game Theory, Nonlinear Optimization Tags , , , ,

In this work, we study stationarity conditions and constraint qualifications (CQs) tailored to Generalized Nash Equilibrium Problems (GNEPs) and analyze their relationships and implications for the global convergence of algorithms. We recall that GNEPs generalize Nash Equilibrium Problems (NEPs) in that the feasible strategy set of each player depends on the strategies chosen by the … Read more

From Computational Certification to Exact Coordinates: Heilbronn’s Triangle Problem on the Unit Square Using Mixed-Integer Optimization

Categories (Mixed) Integer Nonlinear Programming, Combinatorial Optimization, Integer Programming Tags , , , ,

We develop a mixed-integer nonlinear programming (MINLP) approach for the classical Heilbronn triangle problem, demonstrating the capability of modern global optimization solvers to tackle challenging combinatorial geometry problems. A symmetry-breaking strategy based on boundary structure yields a substantially stronger model: for n=9, we compute an epsilon-globally optimal point in 15 minutes on a standard desktop … Read more

Zimpler – Integer Programming, easier

Categories (Mixed) Integer Linear Programming, Modeling Languages and Systems Tags , , , ,

This paper introduces Zimpler, a free tool built on the ZIMPL modeling language to streamline the solution of mixed-integer linear programs (MILP). Zimpler extends existing ZIMPL workflows by integrating native data sources—such as Excel spreadsheets—without requiring manual conversion to text-based tables. In addition, it supports solution refinement by adapting solver outputs into alternative formats, including … Read more

A Multi-Secant Limited-Memory BFGS Method

Categories Unconstrained Optimization Tags ,

We develop multi-secant BFGS-like quasi-Newton updating scheme, which adaptively selects the number of imposed secant conditions and naturally preserves positivity of approximated Hessian. Compact representation and respective limited-memory formulation are also derived. Numerical stability is assured via unconventional damping technique, which symmetrically handles coordinate and gradient differences. Practical relevance of proposed method is demonstrated via … Read more

Water network design and operation optimization: Leveraging approximations

Categories Network Optimization Tags , ,

This study concerns the optimal design and operation of produced water and urban water networks. The optimization formulations of these problems have inherent nonconvexity, making them hard to solve. We address a key source of nonconvexity and difficulty in solving these problems: the representation of frictional pressure changes across network nodes using nonlinear constraints, typically … Read more

Fair Vehicle Routing via Bilevel Optimization

Categories Bilevel Optimization, Transportation Tags , ,

We propose a novel approach to modeling fairness in the Vehicle Routing Problem (VRP) by introducing objective functions based on ordering route lengths, capturing both monotonic and non-monotonic equity measures. Our method ensures allocations that are efficient, capacity-feasible, and equitable according to criteria like min-max, range, Gini, variance, or absolute deviations. To prevent biased or … Read more

Branch-and-Cut for Mixed-Integer Linear Decision-Dependent Robust Optimization

Categories Bilevel Optimization, Branch and Cut Algorithms, Robust Optimization Tags , , , , ,

Decision-dependent robust optimization (DDRO) problems are usually tackled by reformulating them using a strong-duality theorem for the uncertainty set model. If the uncertainty set is, however, described by a mixed-integer linear model, dualization techniques cannot be applied and the literature on solution methods is very scarce. In this paper, we exploit the equivalence of DDRO … Read more

A Modified Projected Gradient Algorithm for Solving Quasiconvex Programming with Applications

Categories Constrained Nonlinear Optimization, Convex Optimization, Nonlinear Optimization Tags , , , ,

In this manuscript, we introduce a novel projected gradient algorithm for solving quasiconvex optimization problems over closed convex sets. The key innovation of our new algorithm is an adaptive, parameter-free stepsize rule that requires no line search and avoids estimating constants, such as Lipschitz modulus. Unlike recent self-adaptive approach given in [17] which typically produce … Read more