Nonconvex Variational Problems Research Articles

On optimality conditions of relaxed non-convex variational problems

Non-convex variational problems in many situations lack a classical solution. Still they can be solved in a generalized sense, e.g., they can be relaxed by means of Young measures. Various sets of optimality conditions of the relaxed non-convex variational problems can be introduced. For example, the so-called “variations” of Young measures lead to a set of optimality conditions, or the Weierstrass maximum principle can be the base of another set of optimality conditions. Moreover the second order necessary and sufficient optimality conditions can be derived from the geometry of the relaxed problem. In this article the sets of optimality conditions are compared. Illustrative examples are included.

A sharp attainment result for nonconvex variational problems

We consider the problem of minimizing autonomous, multiple integrals like $$ \min \left\{ \int_\Omega f\left(u ,\nabla u\right) dx : u\in u_0 + W_0^{1,p}(\Omega) \right\} $$ (℘) where \(f: \ensuremath{\mathbb{R}}\times \ensuremath{\mathbb{R}} ^N\to [0 ,\infty)\) is a continuous, possibly nonconvex function of the gradient variable \(\nabla u\). Assuming that the bipolar function f** of f is affine as a function of the gradient \(\nabla u\) on each connected component of the sections of the detachment set \(\mathcal{D} = \left\{f^{**} < f\right\}\), we prove attainment for (\(\mathcal{P}\)) under mild assumptions on f and f**. We present examples that show that the hypotheses on f and f** considered here for attainment are essentially sharp.

Energy Minimization and Flux Domain Structure in the Intermediate State of a Type-I Superconductor

The intermediate state of a type-I superconductor involves a fine-scale mixture of normal and superconducting domains. We take the viewpoint, due to Landau, that the realizable domain patterns are (local) minima of a nonconvex variational problem. We examine the scaling law of the minimum energy and the qualitative properties of domain patterns achieving that law. Our analysis is restricted to the simplest possible case: a superconducting plate in a transverse magnetic field. Our methods include explicit geometric constructions leading to upper bounds and ansatz-free inequalities leading to lower bounds. The problem is unexpectedly rich when the applied field is near-zero or near-critical. In these regimes there are two small parameters, and the ground state patterns depend on the relation between them.

Image Sharpening by Flows Based on Triple Well Potentials

Image sharpening in the presence of noise is formulated as a non-convex variational problem. The energy functional incorporates a gradient-dependent potential, a convex fidelity criterion and a high order convex regularizing term. The first term attains local minima at zero and some high gradient magnitude, thus forming a triple well-shaped potential (in the one-dimensional case). The energy minimization flow results in sharpening of the dominant edges, while most noisy fluctuations are filtered out.

Adaptive Approximation of Young Measure Solutions in Scalar Nonconvex Variational Problems

This paper addresses the numerical approximation of Young measures appearing as generalized solutions to scalar nonconvex variational problems. We prove a priori and a posteriori error estimates for a macroscopic quantity, the stress. For a scalar three-well problem we show convergence of other quantities such as Young measure support and microstructure region. Numerical experiments indicate that the computational effort in the solution of the large optimization problem is significantly reduced by using an adaptive mesh refinement strategy based on a posteriori error estimates in combination with an active set strategy due to Carstensen and Roubicek [Numer. Math., 84 (2000), pp. 395--414].

The obstacle problem for water tanks

In this paper we discuss the problem of computing and analyzing the static equilibrium of a nonrigid water tank. Specifically, we fix the amount of water contained in the tank, modelled as a membrane. In addition, there are rigid obstacles that constrain the deformation. This amounts to a nonconvex variational problem. We derive the optimality system and its interpretation in terms of equilibrium of forces. A second-order sensitivity analysis, allowing to compute derivatives of solutions and a second-order Taylor expansion of the cost function, is performed, in spite of the fact that the cost function is not twice differentiable. We also study the finite elements discretization, introduce a decomposition algorithm for the numerical computation of the solution, and display numerical results.

On the minimization of some nonconvex double obstacle problems

We consider a nonconvex variational problem for which the set of admissible functions consists of all Lipschitz functions located between two fixed obstacles. It turns out that the value of the minimization problem at hand is equal to zero when the obstacles do not touch each other; otherwise, it might be positive. The results are obtained through the construction of suitable minimizing sequences. Interpolating these minimizing sequences in some discrete space, a numerical analysis is then carried out.

Duality models for some nonclassical problems in the calculus ofvariations

Parametric and nonparametric necessary and sufficient optimality conditions are established for a class of nonconvex variational problems with generalized fractional objective functions and nonlinear inequality constraints containing arbitrary norms. Based on these optimality criteria, ten parametric and parameter‐free dual problems are constructed and appropriate duality theorems are proved. These optimality and duality results contain, as special cases, similar results for minmax fractional variational problems involving square roots of positive semidefinite quadratic forms as well as for variational problems with fractional, discrete max, and conventional objective functions, which are particular cases of the main problem considered in this paper. The duality models presented here subsume various existing duality formulations for variational problems and include variational generalizations of a great variety of cognate dual problems investigated previously in the area of finite‐dimensional nonlinear programming by an assortment of ad hoc methods.

From a Nonlinear, Nonconvex Variational Problem to a Linear, Convex Formulation

Abstract. We propose a general approach to deal with nonlinear, nonconvex variational problems based on a reformulation of the problem resulting in an optimization problem with linear cost functional and convex constraints. As a first step we explicitly explore these ideas to some one-dimensional variational problems and obtain specific conclusions of an analytical and numerical nature.

A reduced theory for thin‐film micromagnetics

AbstractMicromagnetics is a nonlocal, nonconvex variational problem. Its minimizer represents the ground‐state magnetization pattern of a ferromagnetic body under a specified external field. This paper identifies a physically relevant thin‐film limit and shows that the limiting behavior is described by a certain “reduced” variational problem. Our main result is the Γ‐convergence of suitably scaled three‐dimensional micromagnetic problems to a two‐dimensional reduced problem; this implies, in particular, convergence of minimizers for any value of the external field. The reduced problem is degenerate but convex; as a result, it determines some (but not all) features of the ground‐state magnetization pattern in the associated thin‐film limit. © 2002 Wiley Periodicals, Inc.

Error Estimates for the Adaptive Computation of a Scalar Three Well Problem

We investigate the numerical approximation of Young measure solutions appearing as generalised solutions in scalar non-convex variational problems. A priori and a posteriori error estimates for a macroscopic quantity, i.e., the stress, are given. Numerical experiments for a scalar three well problem, occurring as a subproblem in the theory of phase transitions in crystalline solids, show that the computational effort can be significantly reduced using an adaptive mesh-refinement strategy combined with an active set technique by Carstensen and Roubíček.

NUMERICAL SOLUTION OF A CLASS OF NON-CONVEX VARIATIONAL PROBLEMS BY SQP

ABSTRACT Non-convex variational problems can be solved numerically by sequential quadratic programming (SQP), after relaxation by means of Young measures. The approximate solutions obtained by SQP converge weakly to the solution of the relaxed problem. Strong convergence is provided by compact imbedding of the space of feasible mappings due to assumptions on the growth of the involved integrand. An illustrative two-dimensional example with a known exact solution is included.

An existence result for a nonconvex variational problem via regularity

Local Lipschitz continuity of minimizers of certain integrals of the Calculus of Variations is obtained when the integrands are convex with respect to the gradient variable, but are not necessarily uniformly convex . In turn, these regularity results entail existence of minimizers of variational problems with non-homogeneous integrands nonconvex with respect to the gradient variable. The x -dependence, explicitly appearing in the integrands, adds significant technical difficulties in the proof.

Local Stress Regularity in Scalar Nonconvex Variational Problems

In light of applications to relaxed problems in the calculus of variations, this paper addresses convex but not necessarily strictly convex minimization problems. A class of energy functionals is described for which any stress field $\sigma$ in $L^q(\Omega)$ with $\operatorname{{\rm div}}\sigma$ in $W^{1,p'}(\Omega)$ belongs to $ W^{1,q}_{loc}(\Omega)$. The condition on $\operatorname{{\rm div}}\sigma$ holds, for example, for solutions of the Euler--Lagrange equations involving additional lower-order terms. Applications include the scalar double-well potential, an optimal design problem, a vectorial double-well problem in a compatible case, and Hencky elastoplasticity with hardening. If the energy density depends only on the modulus of the gradient, we also show regularity up to the boundary.

Numerical analysis of oscillations in a nonconvex problem related to image selective smoothing

We study some numerical properties of a nonconvex variational problem which arises as the continuous limit of a discrete optimization method designed for the smoothing of images with preservation of discontinuities. The functional that has to be minimized fails to attain a minimum value. Instead, minimizing sequences develop gradient oscillations which allow them to reduce the value of the functional. The oscillations of the gradient exhibit analogies with microstructures in ordered materials. The pattern of the oscillations is analysed numerically by using discrete parametrized measures.

Local regularity of solutions of variational problems for the equilibrium configuration of an incompressible, multiphase elastic body

We consider a multiphase, incompressible, elastic body with k preferred states whose equilibrium conguration is described in terms of a nonconvex variational problem. We pass to a suitable relaxed variational integral whose solution has the meaning of the strain tensor and also study the associated dual problem for the stresses. At rst we show that the strain tensor is smooth near any point of strict J 1-quasiconvexity of the relaxed integrand. Then we use this result to get regularity of the stress tensor on the union of pure phases at least in the two-dimensional case.

Existence, uniqueness and qualitative properties of minima to radially symmetric non-coercive non-convex variational problems

We are concerned with the problem of existence, uniqueness and qualitative properties of solutions to the radially symmetric variational problem \[ \min_{u\in\Wuu(B_R)}\int_{B_R} \pq{f\pt{\mod{x},\mod{\nabla u(x)}}+h(|x|,u(x))}\,dx, \] where $B_R$ is the ball of ${\mathbb R}^n$ centered at the origin and with radius $R>0$ , the map $f\colon [0,R]\times[0,+\infty[\to{\overline{\mathbb R}} $ is a normal integrand, and $h\colon[0,R]\times\R\to\R$ is a convex function of the second variable. This kind of problems, with non-convex lagrangians with respect to $\nabla u$ , arise in various fields of applied sciences, such as optimal design and nonlinear elasticity.

Explicit Solutions of Nonconvex Variational Problems in Dimension One

We propose a method of finding the generalized solutions of nonconvex variational problems by solving an appropriate differential inclusion that is motivated by necessary conditions of optimality for such generalized minimizers.

Numerical approximation of young measuresin non-convex variational problems

Summary. In non-convex optimisation problems, in particular in non-convex variational problems, there usually does not exist any classical solution but only generalised solutions which involve Young measures. In this paper, first a suitable relaxation and approximation theory is developed together with optimality conditions, and then an adaptive scheme is proposed for the efficient numerical treatment. The Young measures solving the approximate problems are usually composed only from a few atoms. This is the main argument our effective active-set type algorithm is based on. The support of those atoms is estimated from the Weierstrass maximum principle which involves a Hamiltonian whose good guess is obtained by a multilevel technique. Numerical experiments are performed in a one-dimensional variational problem and support efficiency of the algorithm.

Domain Branching in Uniaxial Ferromagnets: A Scaling Law for the Minimum Energy

We address the branching of magnetic domains in a uniaxial ferromagnet. Our thesis is that branching is required by energy minimization. To show this, we consider the nonlocal, nonconvex variational problem of micromagnetics. We identify the scaling law of the minimum energy by proving a rigorous lower bound which matches the already-known upper bound. We further show that any domain pattern achieving this scaling law must have average width of order L 2/3, where L is the length of the magnet in the easy direction. Finally we argue that branching is required, by considering the constrained variational problem in which branching is prohibited and the domain structure is invariant in the easy direction. Its scaling law is different.