An interior penalty approach to a large-scale discretized obstacle problem with nonlinear constraints
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An interior penalty approach to a large-scale discretized obstacle problem with nonlinear constraints Jian-Xun Zhao1 · Song Wang2 Received: 2 December 2018 / Accepted: 3 October 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We propose an interior penalty method to solve a nonlinear obstacle problem arising from the discretization of an infinite-dimensional optimization problem. An interior penalty equation is proposed to approximate the mixed nonlinear complementarity problem representing the Karush-Kuhn-Tucker conditions of the obstacle problem. We prove that the penalty equation is uniquely solvable and present a convergence analysis for the solution of the penalty equation when the problem is strictly convex. We also propose a Newton’s algorithm for solving the penalty equation. Numerical experiments are performed to demonstrate the convergence and usefulness of the method when it is used for the two non-trivial test problems. Keywords Interior penalty method · Large-scale obstacle problem · Approximate equation · Mixed nonlinear complementarity problem · Optimization problems with nonlinear constraints Mathematics Subject Classifications (2010) 90C33 · 65K10 · 49M30
Jian-Xun Zhao
[email protected] Song Wang [email protected] 1
School of Mathematics, Tianjin University, Tianjin 300354, People’s Republic of China
2
Shenzhen Audencia Business School, WeBank Institute of Fintech, Guangdong Laboratory of Artificial Intelligence and Digital Economics (SZ), Shenzhen University, Shenzhen, 518060, China
Numerical Algorithms
1 Introduction We consider the following optimization problem in finite dimensions min F (x)
(1.1)
subject to g(x) ≤ 0,
(1.2)
x∈Rn
where n is a positive integer, F : Rn → R is a nonlinear differentiable function, and g(x) = (g1 (x), g2 (x), . . . , gm (x)) with each gj : Rn → R (j = 1, 2, . . . , m) a nonlinear function. This finite-dimensional problem arises from discretization of the following infinite-dimensional nonlinear obstacle problem: which appears naturally in many areas of science, engineering, management, and finance. min F (u)
(1.3)
subject to G (u) ≤ 0,
(1.4)
u∈H
where H is a functional space, F is a functional on H of usually u and its derivatives, and G is a given nonlinear differential operator. This infinite-dimensional problem can be regarded as a nonlinear obstacle problem with a nonlinear obstacle (or constraint), as both F and G are nonlinear differential operators. Numerical methods can be found in the open literature for solving the above problem when G is a linear operator, and techniques for (1.3)–(1.4) when G is nonlinear are very scarce. It is very rare that (1.3)–(1.4) can be solved analytically and numerical approximations to its solution is usually sought in practice. To achieve this, we may use an appropriate technique to discretize (1.3)–(1.4), yielding a finite-dimensional problem of the form (1.1)–(1.2) which needs to be solved numerically. Since n in (1.1)–(1.2) equals roughly the number of mesh no
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