Related papers: Bilevel Imaging Learning Problems as Mathematical …
In this paper, we study the regularity assumptions commonly adopted in bilevel optimization with constrained lower-level problems, including the linear independence constraint qualification, the strict complementary slackness condition, and…
A standard approach to solving optimistic bilevel linear programs (BLPs) is to replace the lower-level problem with its Karush-Kuhn-Tucker (KKT) optimality conditions and reformulate the resulting complementarity constraints using auxiliary…
Integrated learning and optimization (ILO) is a framework in contextual optimization which aims to train a predictive model for the probability distribution of the underlying problem data uncertainty, with the goal of enhancing the quality…
We consider a bilevel optimization approach in function space for the choice of spatially dependent regularization parameters in TV image restoration models. First- and second-order optimality conditions for the bilevel problem are studied,…
In this paper, we study the mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with a parametric generalized equation involving the regular normal cone. We derive a new necessary optimality…
This paper provides necessary and sufficient optimality conditions for abstract constrained mathematical programming problems in locally convex spaces under new qualification conditions. Our approach exploits the geometrical properties of…
Many optimization problems require hyperparameters, i.e., parameters that must be pre-specified in advance, such as regularization parameters and parametric regularizers in variational regularization methods for inverse problems, and…
We consider a smooth pessimistic bilevel optimization problem, where the lower-level problem is convex and satisfies the Slater constraint qualification. These assumptions ensure that the Karush-Kuhn-Tucker (KKT) reformulation of our…
One fundamental problem when solving inverse problems is how to find regularization parameters. This article considers solving this problem using data-driven bilevel optimization, i.e. we consider the adaptive learning of the regularization…
We introduce a unified framework based on bi-level optimization schemes to deal with parameter learning in the context of image processing. The goal is to identify the optimal regularizer within a family depending on a parameter in a…
We present a systematic introduction to first-order optimality conditions for mathematical programs with equilibrium constraints (MPECs), emphasizing the limitations of classical nonlinear programming techniques. The goal is twofold. First,…
Bilevel programming has emerged as a valuable tool for hyperparameter selection, a central concern in machine learning. In a recent study by Ye et al. (2023), a value function-based difference of convex algorithm was introduced to address…
This study explores B-stationarity of mathematical programs with complementarity constraints (MPCCs) and convergence behavior of MPCC algorithms. Special attention is given to the cases with biactive complementarity constraints. First, we…
Second-order optimality conditions of the bilevel programming problems are dependent on the second-order directional derivatives of the value functions or the solution mappings of the lower level problems under some regular conditions,…
In recent years, bilevel approaches have become very popular to efficiently estimate high-dimensional hyperparameters of machine learning models. However, to date, binary parameters are handled by continuous relaxation and rounding…
We consider nonlinear optimization problems with cardinality constraints. Based on a continuous reformulation we introduce second order necessary and sufficient optimality conditions. Under such a second order condition, we can guarantee…
Second-order optimality conditions are essential for nonsmooth optimization, where both the objective and constraint functions are Lipschitz continuous and second-order directionally differentiable. This paper provides no-gap second-order…
Bilevel learning refers to machine learning problems that can be formulated as bilevel optimization models, where decisions are organized in a hierarchical structure. This paradigm has recently gained considerable attention in machine…
Sparsity constrained minimization captures a wide spectrum of applications in both machine learning and signal processing. This class of problems is difficult to solve since it is NP-hard and existing solutions are primarily based on…
We show that a special class of (nonconvex) NMPC problems admits an exact solution by reformulating them as a finite number of convex subproblems, extending previous results to the multi-input case. Our approach is applicable to a special…