Penalty Approximation Research Articles

Penalty approximation method for a double obstacle quasilinear parabolic variational inequality problem

<p style='text-indent:20px;'>In this paper, we first establish a surjectivity result for the sum of a maximal monotone mapping and a generalized pseudomontone mapping by using the generalized Yosida approximation technique. Then, we study a double obstacle quasilinear parabolic variational inequality problem <inline-formula><tex-math id="M1">\begin{document}$ {{\rm{(VIP)}}} $\end{document}</tex-math></inline-formula> by using the surjectivity result and penalty approximation method. In order to deal with the double obstacle constraints, we construct a quasilinear parabolic partial differential penalty equation, then we obtain the solvability of the quasilinear parabolic partial differential penalty equation. Moreover, we show that the set of the solutions to the penalty equation is bounded and every weak cluster point of this set is a solution of the problem <inline-formula><tex-math id="M2">\begin{document}$ {{\rm{(VIP)}}} $\end{document}</tex-math></inline-formula>. At last, as an application, we obtain numerical solutions of two double obstacle parabolic variational inequality problems by using the power penalty approximation method. Through the figures in the examples, we can intuitively see the different numerical solutions of the problems at different times.</p>

Nonconvex multi-view subspace clustering via simultaneously learning the representation tensor and affinity matrix* *This research was supported by the National Natural Science Foundations of China (12071159, U1811464) and the NSF-DMS 1854638 of the United States.

Multi-view subspace clustering, which aims to partition a dataset into its relevant subspaces based on their multi-view features, has been widely applied to identify various characteristics of datasets. The typical model of multi-view subspace clustering in literature often makes use of the nuclear norm to seek the underlying low-rank representation. However, due to the sum property of the singular values defined by tensor nuclear norm, the existing multi-view subspace clustering does not well handle the noise and the illumination variations embedded in multi-view data. To address and improve the robustness and clustering performance, we propose a new nonconvex multi-view subspace clustering model via tensor minimax concave penalty (MCP) approximation associated with rank minimization (NMSC-MCP), which can simultaneously construct the low-rank representation tensor and affinity matrix in a unified framework. Specifically, the nonconvex MCP approximation rank function is adopted to as a tighter tensor rank approximation to discriminate the dimension of features so that better accuracy can be achieved. In addition, we also address the local structure by including both hyper-Laplacian regularization and auto-weighting scheme into the objective function to promote the clustering performance. A corresponding iterative algorithm is then developed to solve the proposed model and the constructed iterative sequence generated by the proposed algorithm is shown to converge to the desirable KKT critical point. Extensive experiments on benchmark datasets have demonstrate the highly desirable effectiveness of our proposed method.

Dynamic Optimal Reinsurance and Dividend Payout in Finite Time Horizon

This paper studies a dynamic optimal reinsurance and dividend-payout problem for an insurance company in a finite time horizon. The goal of the company is to maximize the expected cumulative discounted dividend payouts until bankruptcy or maturity, whichever comes earlier. The company is allowed to buy reinsurance contracts dynamically over the whole time horizon to cede its risk exposure with other reinsurance companies. This is a mixed singular–classical stochastic control problem, and the corresponding Hamilton–Jacobi–Bellman equation is a variational inequality with a fully nonlinear operator and subject to a gradient constraint. We obtain the [Formula: see text] smoothness of the value function and a comparison principle for its gradient function by the penalty approximation method so that one can establish an efficient numerical scheme to compute the value function. We find that the surplus-time space can be divided into three nonoverlapping regions by a risk-magnitude and time-dependent reinsurance barrier and a time-dependent dividend-payout barrier. The insurance company should be exposed to a higher risk as its surplus increases, be exposed to the entire risk once its surplus upward crosses the reinsurance barrier, and pay out all its reserves exceeding the dividend-payout barrier. The estimated localities of these regions are also provided. Funding: This work was supported by the Hong Kong Research Grants Council [Grants GRF 15202421 and GRF 15202817], the Guangdong Basic and Applied Basic Research Foundation [Grants 2021A1515012031 and 2022A1515010263], the National Natural Science Foundation of China [Grants 11901244 and 11971409], the PolyU-SDU Joint Research Center on Financial Mathematics, the CAS AMSS-PolyU Joint Laboratory of Applied Mathematics, and Hong Kong Polytechnic University.

Penalty method for portfolio selection with capital gains tax

AbstractMany finance problems can be formulated as a singular stochastic control problem, where the associated Hamilton‐Jacobi‐Bellman (HJB) equation takes the form of variational inequality, and its penalty approximation equation is linked to a regular control problem. The penalty method, combined with a finite difference scheme, has been widely used to numerically solve singular control problems, and its convergence analysis in literature relies on the uniqueness of solution to the original HJB equation problem. We consider a singular stochastic control problem arising from continuous‐time portfolio selection with capital gains tax, where the associated HJB equation problem admits infinitely many solutions. We show that the penalty method still works and converges to the value function, which is the minimal (viscosity) solution of the HJB equation problem. The key step is to prove that any admissible singular control can be approximated by a sequence of regular controls related to the corresponding penalized equation problem that admits a unique solution. Numerical results are presented to demonstrate the efficiency of the penalty method and to better understand optimal investment strategy in the presence of capital gains tax. Our approach sheds light on the robustness of the penalty method for general singular stochastic control problems.

A power penalty approach to a mixed quasilinear elliptic complementarity problem

In this paper, a power penalty approximation method is proposed for solving a mixed quasilinear elliptic complementarity problem. The mixed complementarity problem is first reformulated as a double obstacle quasilinear elliptic variational inequality problem. A nonlinear elliptic partial differential equation is then defined to approximate the resulting variational inequality by using a power penalty approach. The existence and uniqueness of the solution to the partial differential penalty equation are proved. It is shown that, under some mild assumptions, the sequence of solutions to the penalty equations converges to the unique solution of the variational inequality problem as the penalty parameter tends to infinity. The error estimates of the convergence of this penalty approach are also derived. At last, numerical experimental results are presented to show that the power penalty approximation method is efficient and robust.

Spectral Element Methods a Priori and a Posteriori Error Estimates for Penalized Unilateral Obstacle Problem

The purpose of this paper is the determination of the numerical solution of a classical unilateral stationary elliptic obstacle problem. The numerical technique combines Moreau-Yoshida penalty and spectral finite element approximations. The penalized method transforms the obstacle problem into a family of semilinear partial differential equations. The discretization uses a non-overlapping spectral finite element method with Legendre–Gauss–Lobatto nodal basis using a conforming mesh. The strategy is based on approximating the solution using a spectral finite element method. In addition, by coupling the penalty and the discretization parameters, we prove a priori and a posteriori error estimates where reliability and efficiency of the estimators are shown for Legendre spectral finite element method. Such estimators can be used to construct adaptive methods for obstacle problems. Moreover, numerical results are given to corroborate our error estimates.

Pointwise error estimates for 𝐶⁰ interior penalty approximation of biharmonic problems

The aim of this paper is to derive pointwise global and local best approximation type error estimates for biharmonic problems using the C 0 C^0 interior penalty method. The analysis uses the technique of dyadic decompositions of the domain, which is assumed to be a convex polygon. The proofs require local energy estimates and new pointwise Green’s function estimates for the continuous problem which has independent interest.

Ambient residual penalty approximation of partial differential equations on embedded submanifolds

In this paper, we present a novel approach to the approximate solution of elliptic partial differential equations on compact submanifolds of $\mathbb {R}^{d}$ , particularly compact surfaces and the surface equation ${\Delta }_{\mathbb {M}} u - \lambda u=f$ . In the course of this, we reconsider differential operators on such submanifolds to deduce suitable penalty based functionals. These functionals are based on the residual of the equation in an integral representation, extended by a penalty on the first-order normal derivative. The general framework we develop is accompanied by error analysis and exemplified by numerical examples employing tensor product B-splines.

Penalty Method for Portfolio Selection with Capital Gains Tax

Many finance problems can be formulated as a singular stochastic control problem, where the associated Hamilton-Jacobi-Bellman (HJB) equation takes the form of variational inequality and its penalty approximation equation is linked to a regular control problem. The penalty method, as a finite difference scheme for the penalty equation, has been widely used to numerically solve singular control problems, and its convergence analysis in literature relies on the uniqueness of solution to the original HJB equation problem. We consider a singular stochastic control problem arising from continuous-time portfolio selection with capital gains tax, where the associated HJB equation problem admits infinitely many solutions. We show that the penalty method still works and converges to the value function which is the minimal (viscosity) solution of the HJB equation problem. Numerical results are presented to demonstrate the efficiency of the penalty method and to better understand optimal investment strategy in the presence of capital gains tax. Our approach sheds light on the robustness of the penalty method for general singular stochastic control problems.

Classification and Regression Using an Outer Approximation Projection-Gradient Method

This paper deals with sparse feature selection and grouping for classification and regression. The classification or regression problems under consideration consists of minimizing a convex empirical risk function subject to an $\ell ^1$ constraint, a pairwise $\ell ^\infty$ constraint, or a pairwise $\ell ^1$ constraint. Existing work, such as the Lasso formulation, has focused mainly on Lagrangian penalty approximations, which often require ad hoc or computationally expensive procedures to determine the penalization parameter. We depart from this approach and address the constrained problem directly via a splitting method. The structure of the method is that of the classical gradient-projection algorithm, which alternates a gradient step on the objective and a projection step onto the lower level set modeling the constraint. The novelty of our approach is that the projection step is implemented via an outer approximation scheme in which the constraint set is approximated by a sequence of simple convex sets consisting of the intersection of two half-spaces. Convergence of the iterates generated by the algorithm is established for a general smooth convex minimization problem with inequality constraints. Experiments on both synthetic and biological data show that our method outperforms penalty methods.

The Navier–Stokes Equations Under a Unilateral Boundary Condition of Signorini’s Type

We propose a new outflow boundary condition, a unilateral condition of Signorini’s type, for the incompressible Navier–Stokes equations. The condition is a generalization of the standard free-traction condition. Its variational formulation is given by a variational inequality. We also consider a penalty approximation, a kind of the Robin condition, to deduce a suitable formulation for numerical computations. Under those conditions, we can obtain energy inequalities that are key features for numerical computations. The main contribution of this paper is to establish the well-posedness of the Navier–Stokes equations under those boundary conditions. Particularly, we prove the unique existence of strong solutions of Ladyzhenskaya’s class using the standard Galerkin’s method. For the proof of the existence of pressures, however, we offer a new method of analysis.

A penalty approximation method for a semilinear parabolic double obstacle problem

In this work, we present a novel power penalty method for the approximation of a global solution to a double obstacle complementarity problem involving a semilinear parabolic differential operator and a bounded feasible solution set. We first rewrite the double obstacle complementarity problem as a double obstacle variational inequality problem. Then, we construct a semilinear parabolic partial differential equation (penalized equation) for approximating the variational inequality problem. We prove that the solution to the penalized equation converges to that of the variational inequality problem and obtain a convergence rate that is corresponding to the power used in the formulation of the penalized equation. Numerical results are presented to demonstrate the theoretical findings.

The Effect of Nonsmooth Payoffs on the Penalty Approximation of American Options

This article combines various methods of analysis to draw a comprehensive picture of penalty approximations to the value, hedge ratio, and optimal exercise strategy of American options. We use matched asymptotic expansions to characterize the boundary layers between exercise and hold regions, and to compute first order corrections for representative payoffs on a single asset following a diffusion or jump-diffusion model. Furthermore, we demonstrate how the viscosity theory framework in [E. R. Jakobsen, Asymptot. Anal., 49 (2006), pp. 249--273] can be applied to derive upper and lower bounds on the option value. This analysis confirms the higher order of accuracy in the penalty parameter for convex payoffs (compared to the general case) seen earlier in numerical tests and from asymptotic expansions. In a small extension to [A. Bensoussan and J. L. Lions, Applications of Variational Inequalities in Stochastic Control, Stud. Math. Appl. 12, North-Holland, Amsterdam, New York, Oxford, 1982], we derive weak convergence rates also for option sensitivities for convex payoffs under jump-diffusion models. Finally, we outline applications of the results, including accuracy improvements by extrapolation.

On Riesz minimal energy problems

In Rn, n⩾2, we study the constructive and numerical solution of minimizing the energy relative to the Riesz kernel |x−y|α−n, where 1<α<n, for the Gauss variational problem, considered for finitely many compact, mutually disjoint, boundaryless (n−1)-dimensional C∞-manifolds Γℓ, ℓ∈L, each Γℓ being charged with Borel measures with the sign αℓ≔±1 prescribed. We show that the Gauss variational problem over an affine cone of Borel measures can alternatively be formulated as a minimum problem over an affine cone of surface distributions belonging to the Sobolev–Slobodetski space H−ε/2(Γ), where ε≔α−1 and Γ≔⋃ℓ∈LΓℓ. This allows the application of simple layer boundary integral operators on Γ and, hence, a penalty approximation. A corresponding numerical method is based on the Galerkin–Bubnov discretization with piecewise constant boundary elements. Wavelet matrix compression is applied to sparsify the system matrix. To the discretized problem, a gradient-projection method is applied. Numerical results are presented to illustrate the approach.

Theory of the Space-Time Discontinuous Galerkin Method for Nonstationary Parabolic Problems with Nonlinear Convection and Diffusion

The paper presents the theory of the space-time discontinuous Galerkin finite element method for the discretization of a nonstationary convection-diffusion initial-boundary value problem with nonlinear convection and nonlinear diffusion. The discontinuous Galerkin method is applied separately in space and time using, in general, different space grids on different time levels and different polynomial degrees p and q in space and time discretization. In the space discretization the nonsymmetric, symmetric, and incomplete interior and boundary penalty approximations of diffusion terms are used. The paper is concerned with the analysis of error estimates in the “$L^2(L^2)$”- and “DG”-norm formed by the “$L^2(H^1) $”-seminorm and penalty terms. An important ingredient used in the derivation of the error estimate is the concept of the discrete characteristic function and its properties. In the “DG”-norm the error estimates are optimal with respect to the size of the space grid. They are optimal with respect to the time step if the Dirichlet boundary condition behaves in time as a polynomial of degree $\leq q$. In a general case, the derived estimates become suboptimal in time.

Analysis of space–time discontinuous Galerkin method for nonlinear convection–diffusion problems

The paper presents the theory of the discontinuous Galerkin finite element method for the space–time discretization of a nonstationary convection–diffusion initial-boundary value problem with nonlinear convection and linear diffusion. The problem is not singularly perturbed with dominating convection. The discontinuous Galerkin method is applied separately in space and time using, in general, different space grids on different time levels and different polynomial degrees p and q in space and time dicretization. In the space discretization the nonsymmetric, symmetric and incomplete interior and boundary penalty (NIPG, SIPG, IIPG) approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “L 2(L 2)”- and “DG”-norm formed by the “L 2(H 1)”-seminorm and penalty terms. A special technique based on the use of the Gauss–Radau interpolation and numerical integration has been used for the derivation of an abstract error estimate. In the “DG”-norm the error estimates are optimal with respect to the size of the space grid. They are optimal with respect to the time step, if the Dirichlet boundary condition has behaviour in time as a polynomial of degree ≤ q.

On the numerical solution of minimal energy problems

We study the constructive and numerical solution of minimizing the energy for the Gauss variational problem involving the Newtonian potential. As a special case, we also treat the corresponding condenser problem. These problems are considered for two two-dimensional compact, disjoint Lipschitz manifolds Γ j ⊂ ℝ3, j = 1, 2, charged with measures of opposite sign. Since this minimizing problem over an affine cone of Borel measures with finite Newtonian energy can also be formulated as the minimum problem over an affine cone of surface distributions belonging to the Sobolev–Slobodetski space , which allows the application of simple layer boundary integral operators on Γ, a penalty approximation for the Gauss variational problem can be used. The numerical approximation is based on a Galerkin–Bubnov discretization with piecewise constant boundary elements. To the discretized problem, the projection-iteration is applied where the matrix times vector operations are executed with the fast multipole method. For the condenser problem, we solve the dual problem which reduces in our case to solving two linear boundary integral equations. Here the fast multipole method provides an efficient solution algorithm. We finally present some convergence studies and error estimates.

A dynamic elastic-visco-plastic unilateral contact problem with normal damped response and Coulomb friction

We consider a dynamic frictional contact problem between an elastic-visco-plastic body and a foundation. The contact is modelled with a normal damped response condition of such a type that the normal velocity is restricted with unilateral constraint, associated with the Coulomb law in which the coefficient of friction may depend on the velocity. We derive a variational formulation of the problem which has the form of a system coupling an integro–differential equation for the stress field with an evolutionary variational inequality for the displacement field. This inequality is approximated by a variational equation using a smoothing of the friction and the penalty approximation of the unilateral condition. The existence of a weak solution to the variational equation is proved by the Galerkin method for an auxiliary problem with given viscoplastic part of the stress and a fixed point argument. The solvability of the original problem is proved by passing to the limit of the penalty parameter and the smoothing parameter. This convergence is based on a certain regularity of solutions which is verified with the use of a local rectification of the boundary and a translation method.

A Two-Dimensional Landau-Lifshitz Model in Studying Thin Film Micromagnetics

The paper is concerned with a two-dimensional Landau-Lifshitz equation which was first raised by A. DeSimone and F. Otto, and so fourth, when studying thin film micromagnetics. We get the existence of a local weak solution by approximating it with a higher-order equation. Penalty approximation and semigroup theory are employed to deal with the higher-order equation.

Asymptotic Expansion of Penalty-Gradient Flows in Linear Programming

We establish asymptotic expansions for nonautonomous gradient flows of the form $\dot u(t) = -\nabla f(u(t),r(t))$, where $f(x,r)$ is a penalty approximation of a linear program and the penalty parameter $r(t)$ tends to 0 as $t \to \infty$. Under appropriate conditions we show that every integral curve satisfies $u(t) = u^\infty + r(t)\,d_0^* + \dot r(t)r(t)\,w_0^* + o(\dot r(t)r(t))$ for suitable vectors $u^\infty$, $d_0^*$, and $w_0^*$. We deduce an asymptotic expansion for a related dual trajectory, and we show that the primal-dual limit point is a pair of strictly complementary optimal solutions for the linear program.