non-Lipschitz Term Research Articles

Optimality Conditions, Qualifications and Approximation Method for a Class of Non-Lipschitz Mathematical Programs with Switching Constraints

In this paper, we consider a class of mathematical programs with switching constraints (MPSCs) where the objective involves a non-Lipschitz term. Due to the non-Lipschitz continuity of the objective function, the existing constraint qualifications for local Lipschitz MPSCs are invalid to ensure that necessary conditions hold at the local minimizer. Therefore, we propose some MPSC-tailored qualifications which are related to the constraints and the non-Lipschitz term to ensure that local minimizers satisfy the necessary optimality conditions. Moreover, we study the weak, Mordukhovich, Bouligand, strongly (W-, M-, B-, S-) stationay, analyze what qualifications making local minimizers satisfy the (M-, B-, S-) stationay, and discuss the relationship between the given MPSC-tailored qualifications. Finally, an approximation method for solving the non-Lipschitz MPSCs is given, and we show that the accumulation point of the sequence generated by the approximation method satisfies S-stationary under the second-order necessary condition and MPSC Mangasarian-Fromovitz (MF) qualification.

Mathematical programs with complementarity constraints and a non-Lipschitz objective: optimality and approximation

We consider a class of mathematical programs with complementarity constraints (MPCC) where the objective function involves a non-Lipschitz sparsity-inducing term. Due to the existence of the non-Lipschitz term, existing constraint qualifications for locally Lipschitz MPCC cannot ensure that necessary optimality conditions hold at a local minimizer. In this paper, we present necessary optimality conditions and MPCC-tailored qualifications for the non-Lipschitz MPCC. The proposed qualifications are related to the constraints and the non-Lipschitz term, which ensure that local minimizers satisfy these necessary optimality conditions. Moreover, we present an approximation method for solving the non-Lipschitz MPCC and establish its convergence. Finally, we use numerical examples of sparse solutions of linear complementarity problems and the second-best road pricing problem in transportation science to illustrate the effectiveness of our approximation method for solving the non-Lipschitz MPCC.

Non-Lipschitz Models for Image Restoration with Impulse Noise Removal

Impulse noise is a special type of noise which corrupts a portion of the image pixels while keeping the others unaffected. L$_1$TV, L$_1$Nonconvex, and NonconvexTV are three common variational models for impulse noise removal. However, non-Lipschitz cases were ruled out for L$_1$Nonconvex and NonconvexTV in former works. In this paper, we consider L$_1$non-Lipschitz and non-LipschitzTV. Uniform lower bound theories for these two models are obtained, which yield the models' characteristics. Based on the lower bound theories, we propose two new algorithms for the two models. Instead of introducing auxiliary parameters to convert the non-Lipschitz models into Lipschitz ones in usual approximation methods for non-Lipschitz optimization, the new algorithms overcome the non-Lipschitzian by iteratively adding constraints on the supports of non-Lipschitz terms. The new algorithms can be easily implemented, and the global convergence is also established. Numerical examples are given to show good performance of the algorithm and the rationality of the theories. The advantageous models for difference cases are also discussed.

Necessary optimality conditions and exact penalization for non-Lipschitz nonlinear programs

When the objective function is not locally Lipschitz, constraint qualifications are no longer sufficient for Karush–Kuhn–Tucker (KKT) conditions to hold at a local minimizer, let alone ensuring an exact penalization. In this paper, we extend quasi-normality and relaxed constant positive linear dependence condition to allow the non-Lipschitzness of the objective function and show that they are sufficient for KKT conditions to be necessary for optimality. Moreover, we derive exact penalization results for the following two special cases. When the non-Lipschitz term in the objective function is the sum of a composite function of a separable lower semi-continuous function with a continuous function and an indicator function of a closed subset, we show that a local minimizer of our problem is also a local minimizer of an exact penalization problem under a local error bound condition for a restricted constraint region and a suitable assumption on the outer separable function. When the non-Lipschitz term is the sum of a continuous function and an indicator function of a closed subset, we also show that our problem admits an exact penalization under an extended quasi-normality involving the coderivative of the continuous function.

Non-asymptotic oracle inequalities for the high-dimensional cox regression via lasso

We consider finite sample properties of the regularized high-dimensional Cox regression via lasso. Existing literature focuses on linear models or generalized linear models with Lipschitz loss functions, where the empirical risk functions are the summations of independent and identically distributed (iid) losses. The summands in the negative log partial likelihood function for censored survival data, however, are neither iid nor Lipschitz.We first approximate the negative log partial likelihood function by a sum of iid non-Lipschitz terms, then derive the non-asymptotic oracle inequalities for the lasso penalized Cox regression using pointwise arguments to tackle the difficulties caused by lacking iid Lipschitz losses.

Probabilistic representation of solutions for quasi-linear parabolic PDE via FBSDE with reflecting boundary conditions

A probabilistic representation of the solution (in the viscosity sense) of a quasi-linear parabolic PDE system with non-lipschitz terms and a Neumann boundary condition is given via a fully coupled forward-backward stochastic differential equation with a reflecting term in the forward equation. The extension of previous results consists on the relaxation on the Lipschitz assumption on the drift coefficient of the forward equation, using a previous result of the authors.

The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations

The main objective of this paper is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see’s) and stochastic partial differential equations (spde’s) near stationary solutions. Such characterization is realized through the long-term behavior of the solution field near stationary points. The analysis falls in two parts 1, 2. In Part 1, we prove general existence and compactness theorems for Ck-cocycles of semilinear see’s and spde’s. Our results cover a large class of semilinear see’s as well as certain semilinear spde’s with Lipschitz and non-Lipschitz terms such as stochastic reaction diffusion equations and the stochastic Burgers equation with additive infinite-dimensional noise. In Part 2, stationary solutions are viewed as cocycle-invariant random points in the infinite-dimensional state space. The pathwise local structure of solutions of semilinear see’s and spde’s near stationary solutions is described in terms of the almost sure long-time behavior of trajectories of the equation in relation to the stationary solution. More specifically, we establish local stable manifold theorems for semilinear see’s and spde’s (Theorems 2.4.1-2.4.4). These results give smooth stable and unstable manifolds in the neighborhood of a hyperbolic stationary solution of the underlying stochastic equation. The stable and unstable manifolds are stationary, live in a stationary tubular neighborhood of the stationary solution and are asymptotically invariant under the stochastic semiflow of the see/spde. Furthermore, the local stable and unstable manifolds intersect transversally at the stationary point, and the unstable manifolds has fixed finite dimension. The proof uses infinite-dimensional multiplicative ergodic theory techniques, interpolation and perfection arguments (Theorem 2.2.1).

Optimal Control Problems for Stochastic Reaction-Diffusion Systems with Non-Lipschitz Coefficients

By using the dynamic programming approach, we study a control problem for a class of stochastic reaction-diffusion systems with coefficients having polynomial growth. In the cost functional a non-Lipschitz term appears, and this allows us to treat the quadratic case, which is of interest in the applications. The corresponding Hamilton--Jacobi--Bellman equation is first resolved by a fixed point argument in a small time interval and then is extended to arbitrary time intervals by suitable a priori estimates. The main ingredient in the proof is the smoothing effect of the transition semigroup associated with the uncontrolled system.