arXiv subjects

Jane J. Ye

Publications and source records attributed to Jane J. Ye.

At least 19 recordsLinked to original sources

arXiv2026

Directional Optimality Conditions for Optimization Problems in Asplund Spaces

This paper develops directional necessary optimality conditions for constrained optimization problems in Asplund spaces. Under directional metric subregularity, we first derive a directional optimality condition in terms of limiting subdifferentials. We then introduce sufficient conditions for directional metric subregularity and establish their relationships with directional pseudo-normality and quasi-normality. For systems with joint constraints, we further propose a joint criterion for directional metric subregularity and use it to obtain necessary optimality conditions. The results not only extend several finite-dimensional directional constructions to an Asplund-space setting, but also yield conclusions that are new even in finite dimensions.

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arXiv2026

Directional Subdifferentials of the Value Function in Asplund Spaces

Directional subdifferentials of the value function provide a quantitative measure of optimal value response to perturbations. While existing results are largely limited to finite-dimensional settings, this paper develops a comprehensive variational framework in Asplund spaces. We establish essential directional calculus rules, extending directional nonsmooth analysis to infinite dimensions. To address the lack of compactness of bounded sets in infinite-dimensional spaces, we introduce a new directional condition, under which we derive upper estimates for directional limiting and singular subdifferentials of the value function. These results provide a refined analytical foundation for sensitivity analysis in infinite-dimensional hierarchical systems.

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arXiv2026

Directional first order approach for a class of bilevel programs

In this paper, we study a class of bilevel optimization program, where the feasible set of the lower level program is independent of the upper level variable. For bilevel programs it is known that the first order reformulation of a bilevel program requires the convexity of the lower level program while reformulations involving the value function result in difficult optimization problems. In this paper we propose a directional first order approach which does not require convexity of the lower level program. First, we propose some conditions under which the lower level program can be equivalently characterized by its first order condition over a directional neighborhood around the local optimal condition. Next we give some conditions under which the classical first order optimality condition in the form of M-stationary condition still holds as a necessary optimality condition for the first order reformulation of the bilevel program even when the lower level program is nonconvex.

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arXiv2026

Partial Lagrange Multiplier Expressions and Disjunctive Decompositions for Bilevel Optimization

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 Karush-Kuhn-Tucker (KKT) conditions with a sparse type of Lagrange multipliers. This kind of Lagrange multipliers can be conveniently represented by polynomials, for which we call partial Lagrange multiplier expressions (PLMEs). By doing this, each branch problem of the disjunctive program can be solved efficiently by polynomial optimization techniques. Solving each branch problem either returns infeasibility or gives a candidate local or global optimizer for the original bilevel optimization. We give necessary and sufficient conditions for these candidates to be global optimizers, and sufficient conditions for the local optimality. Numerical experiments are also presented to show the efficiency of the method.

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arXiv2024

Moreau Envelope Based Difference-of-weakly-Convex Reformulation and Algorithm for Bilevel Programs

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 bilevel programs. This approach proves particularly powerful when dealing with scenarios where the lower-level problem exhibits convexity in both the upper-level and lower-level variables. Examples of such scenarios include support vector machines and $\ell_1$ and $\ell_2$ regularized regression. In this paper, we significantly expand the range of applications, now requiring convexity only in the lower-level variables of the lower-level program. We present an innovative single-level difference of weakly convex reformulation based on the Moreau envelope of the lower-level problem. We further develop a sequentially convergent Inexact Proximal Difference of Weakly Convex Algorithm (iP-DwCA). To evaluate the effectiveness of the proposed iP-DwCA, we conduct numerical experiments focused on tuning hyperparameters for kernel support vector machines on simulated data.

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arXiv2023

Calm local optimality for nonconvex-nonconcave minimax problems

Nonconvex-nonconcave minimax problems have found numerous applications in various fields including machine learning. However, questions remain about what is a good surrogate for local minimax optimum and how to characterize the minimax optimality. Recently Jin, Netrapalli, and Jordan (ICML 2020) introduced a concept of local minimax point and derived optimality conditions for the smooth and unconstrained case. In this paper, we introduce the concept of calm local minimax point, which is a local minimax point with a calm radius function. With the extra calmness property we obtain first and second-order sufficient and necessary optimality conditions for a very general class of nonsmooth nonconvex-nonconcave minimax problem. Moreover we show that the calm local minimax optimality and the local minimax optimality coincide under a weak sufficient optimality condition for the maximization problem. This equivalence allows us to derive stronger optimality conditions under weaker assumptions for local minimax optimality.

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arXiv2023

Directional subdifferential of the value function

The directional subdifferential of the value function gives an estimate on how much the optimal value changes under a perturbation in a certain direction. In this paper we derive upper estimates for the directional limiting and singular subdifferential of the value function for a very general parametric optimization problem. We obtain a characterization for the directional Lipschitzness of a locally lower semicontinuous function in terms of the directional subdifferentials. Based on this characterization and the derived upper estimate for the directional singular subdifferential, we are able to obtain a sufficient condition for the directional Lipschitzness of the value function. Finally, we specify these results for various cases when all functions involved are smooth, when the perturbation is additive, when the constraint is independent of the parameter, or when the constraints are equalities and inequalities. Our results extend the corresponding results on the sensitivity of the value function to allow directional perturbations. Even in the case of full perturbations, our results recover or even extend some existing results, including the Danskin's theorem.

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arXiv2023

Necessary and sufficient conditions for multiple objective optimal regression designs

We typically construct optimal designs based on a single objective function. To better capture the breadth of an experiment's goals, we could instead construct a multiple objective optimal design based on multiple objective functions. While algorithms have been developed to find multi-objective optimal designs (e.g. efficiency-constrained and maximin optimal designs), it is far less clear how to verify the optimality of a solution obtained from an algorithm. In this paper, we provide theoretical results characterizing optimality for efficiency-constrained and maximin optimal designs on a discrete design space. We demonstrate how to use our results in conjunction with linear programming algorithms to verify optimality.

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arXiv2023

Relaxed constant positive linear dependence constraint qualification for disjunctive programs

The disjunctive system is a system involving a disjunctive set which is the union of finitely many polyhedral convex sets. In this paper, we introduce a notion of the relaxed constant positive linear dependence constraint qualification (RCPLD) for the disjunctive system. For a disjunctive system, our notion is weaker than the one we introduced for a more general system recently (J. Glob. Optim. 2020) and is still a constraint qualification. To obtain the local error bound for the disjunctive system, we introduce the piecewise RCPLD under which the error bound property holds if all inequality constraint functions are subdifferentially regular and the rest of the constraint functions are smooth. We then specialize our results to the ortho-disjunctive program, which includes the mathematical program with equilibrium constraints (MPEC), the mathematical program with vanishing constraints (MPVC) and the mathematical program with switching constraints (MPSC) as special cases. For MPEC, we recover MPEC-RCPLD, an MPEC variant of RCPLD and propose the MPEC piecewise RCPLD to obtain the {error bound property}. For MPVC, we introduce new constraint qualifications MPVC-RCPLD and the piecewise RCPLD, which also implies the local error bound. For MPSC, we show that both RCPLD and the piecewise RCPLD coincide and hence it leads to the local error bound.

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arXiv2023

Combined approach with second-order optimality conditions for bilevel programming problems

In this paper, we propose a combined approach with second-order optimality conditions of the lower level problem to study constraint qualifications and optimality conditions for bilevel programming problems. The new method is inspired by the combined approach developed by Ye and Zhu in 2010, where the authors combined the classical first-order and the value function approaches to derive new necessary optimality conditions. In our approach, we add a second-order optimality condition to the combined program as a new constraint. We show that when all known approaches fail, adding the second-order optimality condition as a constraint makes the corresponding partial calmness condition and the resulting necessary optimality condition easier to hold. We also give some discussions on advantages and disadvantages of the combined approaches with the first-order and the second-order information.

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arXiv2022

Optimality conditions and constraint qualifications for cardinality constrained optimization problems

The cardinality constrained optimization problem (CCOP) is an optimization problem where the maximum number of nonzero components of any feasible point is bounded. In this paper, we consider CCOP as a mathematical program with disjunctive subspaces constraints (MPDSC). Since a subspace is a special case of a convex polyhedral set, MPDSC is a special case of the mathematical program with disjunctive constraints (MPDC). Using the special structure of subspaces, we are able to obtain more precise formulas for the tangent and (directional) normal cones for the disjunctive set of subspaces. We then obtain first and second order optimality conditions by using the corresponding results from MPDC. Thanks to the special structure of the subspace, we are able to obtain some results for MPDSC that do not hold in general for MPDC. In particular we show that the relaxed constant positive linear dependence (RCPLD) is a sufficient condition for the metric subregularity/error bound property for MPDSC which is not true for MPDC in general. Finally we show that under all constraint qualifications presented in this paper, certain exact penalization holds for CCOP.

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arXiv2022

Difference of convex algorithms for bilevel programs with applications in hyperparameter selection

In this paper, we present difference of convex algorithms for solving bilevel programs in which the upper level objective functions are difference of convex functions, and the lower level programs are fully convex. This nontrivial class of bilevel programs provides a powerful modelling framework for dealing with applications arising from hyperparameter selection in machine learning. Thanks to the full convexity of the lower level program, the value function of the lower level program turns out to be convex and hence the bilevel program can be reformulated as a difference of convex bilevel program. We propose two algorithms for solving the reformulated difference of convex program and show their convergence under very mild assumptions. Finally we conduct numerical experiments to a bilevel model of support vector machine classification.

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arXiv2022

Value Function Based Difference-of-Convex Algorithm for Bilevel Hyperparameter Selection Problems

Gradient-based optimization methods for hyperparameter tuning guarantee theoretical convergence to stationary solutions when for fixed upper-level variable values, the lower level of the bilevel program is strongly convex (LLSC) and smooth (LLS). This condition is not satisfied for bilevel programs arising from tuning hyperparameters in many machine learning algorithms. In this work, we develop a sequentially convergent Value Function based Difference-of-Convex Algorithm with inexactness (VF-iDCA). We show that this algorithm achieves stationary solutions without LLSC and LLS assumptions for bilevel programs from a broad class of hyperparameter tuning applications. Our extensive experiments confirm our theoretical findings and show that the proposed VF-iDCA yields superior performance when applied to tune hyperparameters.

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arXiv2021

Generic property of the partial calmness condition for bilevel programming problems

The partial calmness for the bilevel programming problem (BLPP) is an important condition which ensures that a local optimal solution of BLPP is a local optimal solution of a partially penalized problem where the lower level optimality constraint is moved to the objective function and hence a weaker constraint qualification can be applied. In this paper we propose a sufficient condition in the form of a partial error bound condition which guarantees the partial calmness condition. We analyse the partial calmness for the combined program based on the Bouligand (B-) and the Fritz John (FJ) stationary conditions from a generic point of view. Our main result states that the partial error bound condition for the combined programs based on B and FJ conditions are generic for an important setting with applications in economics and hence the partial calmness for the combined program is not a particularly stringent assumption. Moreover we derive optimality conditions for the combined program for the generic case without any extra constraint qualifications and show the exact equivalence between our optimality condition and the one by Jongen and Shikhman given in implicit form. Our arguments are based on Jongen, Jonker and Twilt's generic (five type) classification of the so-called generalized critical points for one-dimensional parametric optimization problems and Jongen and Shikhman's generic local reductions of BLPPs.

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arXiv2021

Optimality Conditions and Exact Penalty for Mathematical Programs with Switching Constraints

In this paper, we give an overview on optimality conditions and exact penalization for the mathematical program with switching constraints (MPSC). MPSC is a new class of optimization problems which has some important applications. It is well-known that if MPSC is treated as a standard nonlinear program, some of the usual constraint qualifications may fail and to deal with this issue one could reformulate it as a mathematical program with disjunctive constraints (MPDC). In this paper we first survey recent results on constraint qualifications and optimality conditions for MPDC and then apply them to MPSC to obtain the corresponding constraint qualifications and optimality conditions. Moreover we provide two types of sufficient conditions for the local error bound and exact penalty results for MPSC. One comes from the directional quasi-normality for MPDC and the other is obtained by using the local decomposition approach.

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arXiv2020

Relaxed constant positive linear dependence constraint qualification and its application to bilevel programs

Relaxed constant positive linear dependence constraint qualification (RCPLD) for a system of smooth equalities and inequalities is a constraint qualification that is weaker than the usual constraint qualifications such as Mangasarian Fromovitz constraint qualification and the linear constraint qualification. Moreover RCPLD is known to induce an error bound property. In this paper we extend RCPLD to a very general feasibility system which may include Lipschitz continuous inequality constraints, complementarity constraints and abstract constraints. We show that this RCPLD for the general system is a constraint qualification for the optimality condition in terms of limiting subdifferential and limiting normal cone and it is a sufficient condition for the error bound property under the strict complementarity condition for the complementarity system and Clarke regularity conditions for the inequality constraints and the abstract constraint set. Moreover we introduce and study some sufficient conditions for RCPLD including the relaxed constant rank constraint qualification (RCRCQ). Finally we apply our results to the bilevel program.

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arXiv2019

Constraint qualifications and optimality conditions in bilevel optimization

In this paper we study constraint qualifications and optimality conditions for bilevel programming problems. We strive to derive checkable constraint qualifications in terms of problem data and applicable optimality conditions. For the bilevel program with convex lower level program we discuss drawbacks of reformulating a bilevel programming problem by the mathematical program with complementarity constraints and present a new sharp necessary optimality condition for the reformulation by the mathematical program with a generalized equation constraint. For the bilevel program with a nonconvex lower level program we propose a relaxed constant positive linear dependence (RCPLD) condition for the combined program.

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arXiv2019

Exact formula for the second-order tangent set of the second-order cone complementarity set

The second-order tangent set is an important concept in describing the curvature of the set involved. Due to the existence of the complementarity condition, the second-order cone (SOC) complementarity set is a nonconvex set. Moreover, unlike the vector complementarity set, the SOC complementarity set is not even the union of finitely many polyhedral convex sets. Despite these difficulties, we succeed in showing that like the vector complementarity set, the SOC complementarity set is second-order directionally differentiable and an exact formula for the second-order tangent set of the SOC complementarity set can be given. We derive these results by establishing the relationship between the second-order tangent set of the SOC complementarity set and the second-order directional derivative of the projection operator over the second-order cone, and calculating the second-order directional derivative of the projection operator over the second-order cone. As an application, we derive second-order necessary optimality conditions for the mathematical program with second-order cone complementarity constraints.

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