Nonsmooth Version Research Articles

Path differentiability of ODE flows

We consider flows of ordinary differential equations (ODEs) driven by path differentiable vector fields. Path differentiable functions constitute a proper subclass of Lipschitz functions which admit conservative gradients, a notion of generalized derivative compatible with basic calculus rules. Our main result states that such flows inherit the path differentiability property of the driving vector field. We show indeed that forward propagation of derivatives given by the sensitivity differential inclusions provides a conservative Jacobian for the flow. This allows to propose a nonsmooth version of the adjoint method, which can be applied to integral costs under an ODE constraint. This result constitutes a theoretical ground to the application of small step first order methods to solve a broad class of nonsmooth optimization problems with parameterized ODE constraints. This is illustrated with the convergence of small step first order methods based on the proposed nonsmooth adjoint.

Generalized Fountain Theorem for Locally Lipschitz Functionals and application

In this paper, we obtain a nonsmooth version of the infinite-dimensional Fountain Theorem established by Batkam and Colin (2013). No symmetry condition on the energy functional is needed in our formulation. As an application, we prove the existence of multiple solutions for the following class of elliptic system (S)Δu−u∈[f̲(x,u,v),f¯(x,u,v)]a.e inRN−Δv+v∈[g̲(x,u,v),g¯(x,u,v)]a.e inRN,u,v∈H1(RN), where f and g are measurable functions that satisfy some technical conditions.

Asymptotic convergence of solutions to the forest kinematic model

We study the asymptotic behavior of global solutions to forest kinetic model equations composed of young trees, old trees, and air-borne seeds. Under some parameter assumptions, we prove the asymptotic convergence of global solutions to a stationary solution. To this end, we show a non-smooth version of the Łojasiewicz–Simon gradient inequality on a suitable functional space and a certain norm estimate of the time derivative of global solutions.

Lagrange multiplier characterizations of constrained best approximation with nonsmooth nonconvex constraints

In this paper, we consider the constraint set K := {x ? Rn : gj(x)? 0,? j = 1,2,...,m} of inequalities with nonsmooth nonconvex constraint functions gj : Rn ? R (j = 1,2,...,m).We show that under Abadie?s constraint qualification the ?perturbation property? of the best approximation to any x in Rn from a convex set ?K := C ? K is characterized by the strong conical hull intersection property (strong CHIP) of C and K, where C is an arbitrary non-empty closed convex subset of Rn: By using the idea of tangential subdifferential and a non-smooth version of Abadie?s constraint qualification, we do this by first proving a dual cone characterization of the constraint set K. Moreover, we present sufficient conditions for which the strong CHIP property holds. In particular, when the set ?K is closed and convex, we show that the Lagrange multiplier characterizations of constrained best approximation holds under a non-smooth version of Abadie?s constraint qualification. The obtained results extend many corresponding results in the context of constrained best approximation. Several examples are provided to clarify the results.

Necessary and sufficient optimality conditions using convexifactors for mathematical programs with equilibrium constraints

The main aim of this paper is to develop necessary Optimality conditions using Convexifactors for mathematical programs with equilibrium constraints (MPEC). For this purpose a nonsmooth version of the standard Guignard constraint qualification (GCQ) and strong stationarity are introduced in terms of convexifactors for MPEC. It is shown that Strong stationarity is the first order necessary optimality condition under nonsmooth version of the standard GCQ. Finally, notions of asymptotic pseudoconvexity and asymptotic quasiconvexity are used to establish the sufficient optimality conditions for MPEC.

A new variational inequality in the calculus of variations and Lipschitz regularity of minimizers

We consider a problem of the calculus of variations of the form(P){MinimizeI(x):=∫abΛ(t,x(t),x′(t))dt+Ψ(x(a),x(b))Subject to:x∈W1,m([a,b];Rn),x′(t)∈C a.e., x(t)∈Σ∀t∈[a,b], where Λ:[a,b]×Rn×Rn→R∪{+∞} is Borel measurable, C is a cone, Σ is a nonempty subset of Rn and Ψ is an arbitrary extended valued function. We prove that if t↦Λ(t,x,ξ) satisfies a nonsmooth version of Cesari's Condition (S), then any local minimizer of (P) in the AC norm is Lipschitz whenever Λ is coercive. The proof is obtained via a new variational inequality formulated here, that holds under the extended Condition (S) (just Borel measurability if Λ is autonomous): For a.e. t in [a,b],(W)Λ(t,x⁎(t),x⁎′(t)v)v−Λ(t,x⁎(t),x⁎′(t))≥p(t)(v−1),∀v>0, where p(t) is an absolutely continuous arc, the derivative of which belongs to a suitable subdifferential of Λ(⋅,x⁎(t),x⁎′(t)) a.e. on [a,b]. The proof of the directional Weierstrass type condition (W) is based on Clarke's nonsmooth recent versions of the Maximum Principle.

Some hemivariational inequalities in the Euclidean space

AbstractThe purpose of this paper is to study the existence of weak solutions for some classes of hemivariational problems in the Euclidean space ℝd(d≥ 3). These hemivariational inequalities have a variational structure and, thanks to this, we are able to find a non-trivial weak solution for them by using variational methods and a non-smooth version of the Palais principle of symmetric criticality for locally Lipschitz continuous functionals, due to Krawcewicz and Marzantowicz. The main tools in our approach are based on appropriate theoretical arguments on suitable subgroups of the orthogonal groupO(d) and their actions on the Sobolev spaceH1(ℝd). Moreover, under an additional hypotheses on the dimensiondand in the presence of symmetry on the nonlinear datum, the existence of multiple pairs of sign-changing solutions with different symmetries structure has been proved. In connection to classical Schrödinger equations a concrete and meaningful example of an application is presented.

Existence and multiplicity of nontrivial solutions of biharmonic equations via differential inclusion

<p style='text-indent:20px;'>This paper is concerned with a class of biharmonic elliptic differential inclusion in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb R^N $\end{document}</tex-math></inline-formula>. Under various growth conditions on the nonlinearity, new results on existence and multiplicity of solutions are derived. The main tools used in this paper are the nonsmooth version of mountain pass theorem and the generalized gradient for locally Lipschitz functionals.

Fixed Points of Non-Smooth Functions on Finite Dimensional Ordered Banach Spaces via Clarke Generalized Jacobian

Considering Lipschitz functions which are not necessarily Fr´echet differentiable, we obtain a non-smooth version of Lakshmikantham’s theorem in finite dimensional ordered Banach spaces . We also present an application of the obtained result in dynamical Coulomb friction problem.

Pull-Back of Metric Currents and Homological Boundedness of BLD-Elliptic Spaces

Abstract Using the duality of metric currents and polylipschitz forms, we show that a BLD-mapping f : X → Y between oriented cohomology manifolds X and Y induces a pull-back operator f * : M k,loc(Y) → M k,loc(X) between the spaces of metric k-currents of locally finite mass. For proper maps, the pull-back is a right-inverse (up to multiplicity) of the push-forward f * : M k,loc(X) → M k,loc(Y). As an application we obtain a non-smooth version of the cohomological boundedness theorem of Bonk and Heinonen for locally Lipschitz contractible cohomology n-manifolds X admitting a BLD-mapping ℝ n → X.

Three Solutions for Fourth-Order Impulsive Differential Inclusions via Nonsmooth Critical Point Theory

An existence of at least three solutions for a fourth-order impulsive differential inclusion will be obtained by applying a nonsmooth version of a three-critical-point theorem. Our results generalize and improve some known results.

A simheuristic approach for the two-dimensional vehicle routing problem with stochastic travel times

The two-dimensional vehicle routing problem (2L-VRP) is a realistic extension of the classical vehicle routing problem in which customers’ demands are composed by sets of non-stackable items. Examples can be found in real-life applications such as the transportation of furniture or industrial machinery. Often, it is necessary to consider stochastic travel times due to traffic conditions or customers availability. However, there is a lack of works discussing stochastic versions of the 2L-VRP. This paper offers a model of the 2L-VRP with stochastic travel times that also includes penalty costs generated by overtime. To solve this stochastic and non-smooth version of the 2L-VRP, a hybrid simheuristic algorithm is proposed. Our approach combines Monte Carlo simulation, an iterated local search framework, and biased-randomised routing and packing heuristics. Our algorithm is tested on an extensive benchmark, which extends the deterministic one for the 2L-VRP with unrestricted and non-oriented loading.

Trajectories of Differential Inclusions on Riemannian Manifolds and Invariance

We study the trajectories of differential inclusions on complete Riemannian manifolds. The topological properties of the reachable set and invariance of trajectories with respect to a closed set are investigated. In order to use required geometric tools, we introduce a linear growth condition by using the distance function on Riemannian manifold. This condition provides us a compact set containing all trajectories with the same initial value and end time. Finally, we apply our results to study a nonsmooth version of control affine systems, where the vector fields are only continuous or locally Lipschitz.

Convexificators and boundedness of the Kuhn–Tucker multipliers set

In this paper we consider a nonsmooth optimization problem with equality, inequality and set constraints. We propose new constraint qualifications and Kuhn–Tucker type necessary optimality conditions for this problem involving locally Lipschitz functions. The main tool of our approach is the notion of convexificators. We introduce a nonsmooth version of the Mangasarian–Fromovitz constraint qualification and show that this constraint qualification is necessary and sufficient for the Kuhn–Tucker multipliers set to be nonempty and bounded.

Differentiable monotonicity-preserving schemes for discontinuous Galerkin methods on arbitrary meshes

This work is devoted to the design of interior penalty discontinuous Galerkin (dG) schemes that preserve maximum principles at the discrete level for the steady transport and convection–diffusion problems and the respective transient problems with implicit time integration. Monotonic schemes that combine explicit time stepping with dG space discretization are very common, but the design of such schemes for implicit time stepping is rare, and it had only been attained so far for 1D problems. The proposed scheme is based on a piecewise linear dG discretization supplemented with an artificial diffusion that linearly depends on a shock detector that identifies the troublesome areas. In order to define the new shock detector, we have introduced the concept of discrete local extrema. The diffusion operator is a graph-Laplacian, instead of the more common finite element discretization of the Laplacian operator, which is essential to keep monotonicity on general meshes and in multi-dimension. The resulting nonlinear stabilization is non-smooth and nonlinear solvers can fail to converge. As a result, we propose a smoothed (twice differentiable) version of the nonlinear stabilization, which allows us to use Newton with line search nonlinear solvers and dramatically improve nonlinear convergence. A theoretical numerical analysis of the proposed schemes shows that they satisfy the desired monotonicity properties. Further, the resulting operator is Lipschitz continuous and there exists at least one solution of the discrete problem, even in the non-smooth version. We provide a set of numerical results to support our findings.

Optimal Control Policy Tuning and Implementation for a Hybrid Electric Race Car

The Formula 1 car is a state-of-the-art hybrid electric vehicle. Its power unit is composed by a turbocharged gasoline engine and two electric motor/generator units connected to the traction system and to the turbocharger. Such a powertrain offers an additional degree of freedom and therefore requires an energy management system. This supervisory controller has a significant influence on the vehicle’s fuel consumption and on the achievable lap-time. Therefore, a thorough, systematic optimization of the energy management system is a crucial prerequisite to win a race. The complexity of the system and the strict regulations make the time-optimal energy management problem non-trivial to solve and an effective implementation of its solution on the car difficult to achieve. In Ebbesen, S. et al. (2016) and Salazar, M. et al. (2016) a convex numerical solver was designed and the optimal strategies were derived analytically using a non-smooth version of Pontryagin’s Minimum Principle, respectively. Building on these results, we design a nonlinear program to tune an efficient version of the optimal control policy, in order to precisely match various boundary conditions on the fuel consumption and on the battery usage. This allows a simple and robust implementation of the time-optimal control strategies on the vehicle, and guarantees compatibility with the FIA rules. A simulator is then used to test the obtained feedforward controls on one race lap in Barcelona. The results stand comparison with the optimal solutions obtained with the numerical solver and validate the effectiveness of this strategy.

Three solutions for second-order impulsive differential inclusions with Sturm-Liouville boundary conditions via nonsmooth critical point theory

A second-order impulsive differential inclusion with Sturm-Liouville boundary conditions is studied. By using a nonsmooth version of a three critical point theorem of Ricceri, the existence of three solutions is obtained.

Semiparametric Accelerated Failure Time Modeling for Clustered Failure Times From Stratified Sampling

Clustered failure times often arise from studies with stratified sampling designs where it is desired to reduce both cost and sampling error. Semiparametric accelerated failure time (AFT) models have not been used as frequently as Cox relative risk models in such settings due to lack of efficient and reliable computing routines for inferences. The challenge roots in the nonsmoothness of the rank-based estimating functions, and for clustered data, the asymptotic properties of the estimator from the weighted version have not been available. The recently proposed induced smoothing approach, which provides fast and accurate rank-based inferences for AFT models, is generalized to incorporate weights to accommodate stratified sampling designs. The estimator from the induced smoothing weighted estimating equations are shown to be consistent and have the same asymptotic distribution as that from the nonsmooth version, which has not been developed before. The variance of the estimator is estimated by computationally efficient sandwich estimators aided by a multiplier bootstrap. The proposed method is assessed in extensive simulation studies where the estimators appear to provide valid and efficient inferences. A stratified case-cohort design with clustered times to tooth extraction in a dental study illustrates the usefulness of the method.

ON EXISTENCE OF THREE SOLUTIONS FOR $p(x)$-KIRCHHOFF TYPE DIFFERENTIAL INCLUSION PROBLEM VIA NONSMOOTH CRITICAL POINT THEORY

In this paper, we study a class of differential inclusion problems driven by the $p(x)$-Kirchhoff with non-standard growth depending on a real parameter. Working within the framework of variable exponent spaces, a new existence result of at least three solutions for the considered problem is established by using the nonsmooth version three critical points theorem.

Generalized homogeneous systems with applications to nonlinear control: A survey

This survey provides a unified homogeneous perspective on recent advances in the global stabilization of various nonlinear systems with uncertainty. We first review definitions and properties of homogeneous systems and illustrate how the homogeneous system theory can yield elegant feedback stabilizers for certain homogeneous systems. By taking advantage of homogeneity, we then present the so-called Adding a Power Integrator (AAPI) technique and discuss how it can be employed to recursively construct smooth state feedback stabilizers for uncertain nonlinear systems with uncontrollable linearizations. Based on the AAPI technique, a non-smooth version as well as a generalized version of AAPI approaches can be further developed from a homogeneous viewpoint, resulting in solutions to the global stabilization of genuinely nonlinear systems that may not be controlled, even locally, by any smooth state feedback. In the case of output feedback control, we demonstrate in this survey why the homogeneity is the key in developing a homogeneous domination approach, which has been successful in solving some difficult nonlinear control problems including, for instance, the global stabilization of systems with higher-order nonlinearities via output feedback. Finally, we show how the notion of Homogeneity with Monotone Degrees (HWMD) plays a decisive role in unifying smooth and non-smooth AAPI methods under one framework. Other applications of HWMD will be also summarized and discussed in this paper, along the directions of constructing smooth stabilizers for nonlinear systems in special forms and ``low-gain'' controllers for a class of general upper-triangular systems.