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Necessary conditions for high-order optimality in smooth nonlinear constrained optimization are explored and their inherent intricacy discussed. A two-phase minimization algorithm is proposed which can achieve approximate first-, second-…
Minimax optimization problems have attracted significant attention in recent years due to their widespread application in numerous machine learning models. To solve the minimax problem, a wide variety of stochastic optimization methods have…
In this work, we propose different formulations and gradient-based algorithms for deterministic and stochastic bilevel problems with conflicting objectives in the lower level. Such problems have received little attention in the…
This paper studies bilevel polynomial optimization in which lower-level constraint functions depend linearly on lower-level variables. We show that such bilevel program can be reformulated as a disjunctive program by using…
Optimality conditions are central to analysis of optimization problems, characterizing necessary criteria for local minima. Formalizing the optimality conditions within the type-theory-based proof assistant Lean4 provides a precise, robust,…
Learning processes are useful methodologies able to improve knowledge of real phenomena. These are often dependent on hyperparameters, variables set before the training process and regulating the learning procedure. Hyperparameters…
In this paper, we obtain necessary optimality conditions for neural network approximation. We consider neural networks in Manhattan ($l_1$ norm) and Chebyshev ($\max$ norm). The optimality conditions are based on neural networks with at…
This paper addresses the class of continuous-time nonlinear programming problems with equality and inequality constraints. The paper presents necessary optimality conditions of the sequential form. To be more precise, a sequence of…
We propose a new approach to solving bilevel optimization problems, intermediate between solving full-system optimality conditions with a Newton-type approach, and treating the inner problem as an implicit function. The overall idea is to…
Minimax optimization problems are an important class of optimization problems arising from both modern machine learning and from traditional research areas. We focus on the stability of constrained minimax optimization problems based on the…
Bilevel optimization problems, which are problems where two optimization problems are nested, have more and more applications in machine learning. In many practical cases, the upper and the lower objectives correspond to empirical risk…
The Karush-Kuhn-Tucker and value function (lower-level value function, to be precise) reformulations are the most common single-level transformations of the bilevel optimization problem. So far, these reformulations have either been studied…
In this paper, we derive first and second-order optimality conditions of KKT type for locally optimal solutions to a class of multiobjective optimal control problems with endpoint constraint and mixed pointwise constraints. We give some…
Minimax optimization has been central in addressing various applications in machine learning, game theory, and control theory. Prior literature has thus far mainly focused on studying such problems in the continuous domain, e.g.,…
This paper firstly proposes a convex bilevel optimization paradigm to formulate and optimize popular learning and vision problems in real-world scenarios. Different from conventional approaches, which directly design their iteration schemes…
Recently, accelerated algorithms using the anchoring mechanism for minimax optimization and fixed-point problems have been proposed, and matching complexity lower bounds establish their optimality. In this work, we present the surprising…
Binary optimization is a central problem in mathematical optimization and its applications are abundant. To solve this problem, we propose a new class of continuous optimization techniques which is based on Mathematical Programming with…
We present a unified study of first and second order necessary and sufficient optimality conditions for minimax and Chebyshev optimisation problems with cone constraints. First order optimality conditions for such problems can be formulated…
Bilevel optimization has been developed for many machine learning tasks with large-scale and high-dimensional data. This paper considers a constrained bilevel optimization problem, where the lower-level optimization problem is convex with…
Second-order optimality conditions for vector nonlinear programming problems with inequality constraints are studied in this paper. We introduce a new second-order constraint qualification, which includes Mangasarian-Fromovitz constraint…