Additively separable duality theory

In duality theory, there is a trade-off between generality and tractability. Thus, the generality of the Tind-Wolsey framework comes at the expense of an infinite-dimensional dual solution space, even if the primal solution space is finite dimensional. Therefore, the challenge is to impose additional structure on the dual solution space and to identify conditions on the primal program, such that the properties that are typically associated with duality, like weak and strong duality, are preserved.In this paper, we consider real-valuedness, continuity, and additive separability as such additional structures. The virtue of the latter property is that it restores the one-to-one correspondence between primal constraints and dual variables as it exists in Lagrangian duality. The main result of this paper is that, roughly speaking, the existence of realvalued, continuous, and additively separable dual solutions that preserve strong duality is guaranteed, once the primal program satisfies a certain stability condition. The latter condition is ensured by the well-known regularity conditions that imply constraint qualification in Karush-Kuhn-Tucker points. On the other hand, if instead of additive separability, a mild tractability condition is imposed on the dual solution space, then stability turns out to be a necessary condition for strong duality in a well-defined sense. This result, combined with the observation that applicability of some well-known augmented Lagrangian methods to constrained optimization.This study was supported by the Netherlands Foundation for Mathematics (SMC) with financial aid from the Netherlands Organization for Scientific Research (NWO).     

Duality algorithms for nonconvex variational principles

This paper describes, and analyzes, a method of successive approximations for finding critical points of a function which can be written as the difference of two convex functions. The method is based on using a non-convex duality theory. At each iteration one solves a convex, optimization problem. This alternates between the primal and the dual variables. Under very general structural conditions on the problem, we prove that the resulting sequence is a descent sequence, which converges to a critical point of the problem. To illustrate the method, it is applied to some weighted eigenvalue problems, to a problem from astrophysics, and to some semilinear elliptic equations.     

Testing copositivity with the help of difference-of-convex optimization

We consider the problem of minimizing an indefinite quadratic form over the nonnegative orthant, or equivalently, the problem of deciding whether a symmetric matrix is copositive. We formulate the problem as a difference of convex functions problem. Using conjugate duality, we show that there is a one-to-one correspondence between their respective critical points and minima. We then apply a subgradient algorithm to approximate those critical points and obtain an efficient heuristic to verify non-copositivity of a matrix.     

Nonconvex energy functions,related eigenvalue hemivariational inequalities on the sphere and applications

The present paper deals with a new type of eigenvalue problems arising in problems involving nonconvex nonsmooth energy functions. They lead to the search of critical points (e.g. local minima) for nonconvex nonsmooth potential functions which in turn give rise to hemivariational inequalities. For this type of variational expressions the eigenvalue problem is studied here concerning the existence and multiplicity of solutions by applying a critical point theory appropriate for nonsmooth nonconvex functionals.     

Stability of critical shapes for the drag minimization problem in Stokes flow

We study the stability of some critical (or equilibrium) shapes in the minimization problem of the energy dissipated by a fluid (i.e. the drag minimization problem) governed by the Stokes equations. We first compute the shape derivative up to the second order, then provide a sufficient condition for the shape Hessian of the energy functional to be coercive at a critical shape. Under this condition, the existence of such a local strict minimum is then proved using a precise upper bound for the variations of the second order shape derivative of the functional with respect to the coercivity and differentiability norms. Finally, for smooth domains, a lower bound of the variations of the drag is obtained in terms of the measure of the symmetric difference of domains.     

Static duality and a stationary-action application

Conservative dynamical systems propagate as stationary points of the action functional. Using this representation, it has previously been demonstrated that one may obtain fundamental solutions for two-point boundary value problems for some classes of conservative systems via solution of an associated dynamic program. Further, such a fundamental solution may be represented as a set of solutions of differential Riccati equations (DREs), where the solutions may need to be propagated past escape times. Notions of “static duality” and “stat-quad duality” are developed, where the relationship between the two is loosely analogous to that between convex and semiconvex duality. Static duality is useful for smooth functionals where one may not be guaranteed of convexity or concavity. Some simple properties of this duality are examined, particularly commutativity. Application to stationary action is considered, which leads to propagation of DREs past escape times via propagation of stat-quad dual DREs.     

Relative Equilibria of the (1+N)-Vortex Problem

We examine existence and stability of relative equilibria of the n-vortex problem specialized to the case where N vortices have small and equal circulation and one vortex has large circulation. As the small circulation tends to zero, the weak vortices tend to a circle centered on the strong vortex. A special potential function of this limiting problem can be used to characterize orbits and stability. Whenever a critical point of this function is nondegenerate, we prove that the orbit can be continued via the Implicit Function Theorem, and its linear stability is determined by the eigenvalues of the Hessian matrix of the potential. For N≥3 there are at least three distinct families of critical points associated to the limiting problem. Assuming nondegeneracy, one of these families continues to a linearly stable class of relative equilibria with small and large circulation of the same sign. This class becomes unstable as the small circulation passes through zero and changes sign. Another family of critical points which is always nondegenerate continues to a configuration with small vortices arranged in an N-gon about the strong central vortex. This class of relative equilibria is linearly unstable regardless of the sign of the small circulation when N≥4. Numerical results suggest that the third family of critical points of the limiting problem also continues to a linearly unstable class of solutions of the full problem independent of the sign of the small circulation. Thus there is evidence that linearly stable relative equilibria exist when the large and small circulation strengths are of the same sign, but that no such solutions exist when they have opposite signs. The results of this paper are in contrast to those of the analogous celestial mechanics problem, for which the N-gon is the only relative equilibrium for N sufficiently large, and is linearly stable if and only if N≥7.     

Surface water waves as saddle points of the energy

By applying the mountain-pass lemma to an energy functional, we establish the existence of two-dimensional water waves on the surface of an infinitely deep ocean in a constant gravity field. The formulation used, which is due to K. I. Babenko [3, 4] (and later to others, independently), has as its independent variable an amplitude function which gives the surface elevation. Its nonlinear term is purely quadratic but it is nonlocal because it involves the Hilbert transform. Moreover the energy functional from which it is derived is rather degenerate and offers an important challenge in the calculus of variations. In the present treatment the first step is to truncate the integrand, and then to penalize and regularize it. The mountain-pass lemma gives the existence of critical points of the resulting problem. To check that, in the limit of vanishing regularization, the critical points converge to a non-trivial water wave, we need a priori estimates and information on their Morseindex in the spirit of the work by Amann and Zehnder [1] (see also [14]). The amplitudes of the waves so obtained are compared with those obtained from the bifurcation argument of Babenko, and are found to extend the parameter range where existence is known by analytical methods. We also compare our approach with the minimization-under-constraint method used by R. E. L. Turner [25].     

离散共振问题的非平凡解

研究了一类二阶非线性差分方程两点边值问题非平凡解的存在性.假设该问题在无穷远点及零点处均是共振的,利用变分方法,同时考虑正、负能量泛函的临界点,在一定的假设条件下,通过临界群的计算,证明了该问题至少存在一个非平凡解.     

Global Carleman Estimates for Waves and Applications

In this article, we extensively develop Carleman estimates for the wave equation and give some applications. We focus on the case of an observation of the flux on a part of the boundary satisfying the Gamma conditions of Lions. We will then consider two applications. The first one deals with the exact controllability problem for the wave equation with potential. Following the duality method proposed by Fursikov and Imanuvilov in the context of parabolic equations, we propose a constructive method to derive controls that weakly depend on the potentials. The second application concerns an inverse problem for the waves that consists in recovering an unknown time-independent potential from a single measurement of the flux. In that context, our approach does not yield any new stability result, but proposes a constructive algorithm to rebuild the potential. In both cases, the main idea is to introduce weighted functionals that contain the Carleman weights and then to take advantage of the freedom on the Carleman parameters to limit the influences of the potentials.     

Investment under duality risk measure

One index satisfies the duality axiom if one agent, who is uniformly more risk-averse than another, accepts a gamble, the latter accepts any less risky gamble under the index. Aumann and Serrano (2008) show that only one index defined for so-called gambles satisfies the duality and positive homogeneity axioms. We call it a duality index. This paper extends the definition of duality index to all outcomes including all gambles, and considers a portfolio selection problem in a complete market, in which the agent’s target is to minimize the index of the utility of the relative investment outcome. By linking this problem to a series of Merton’s optimum consumption-like problems, the optimal solution is explicitly derived. It is shown that if the prior benchmark level is too high (which can be verified), then the investment risk will be beyond any agent’s risk tolerance. If the benchmark level is reasonable, then the optimal solution will be the same as that of one of the Merton’s series problems, but with a particular value of absolute risk aversion, which is given by an explicit algebraic equation as a part of the optimal solution. According to our result, it is riskier to achieve the same surplus profit in a stable market than in a less-stable market, which is consistent with the common financial intuition.     

Stability estimates for an inverse problem for the Schrödinger equation at negative energy in two dimensions

We study the inverse problem of determining a real-valued potential in the two-dimensional Schrödinger equation at negative energy from the Dirichlet-to-Neumann map. It is known that the problem is ill-posed and a stability estimate of logarithmic type holds. In this article, we prove three new stability estimates. The main feature of the first one is that the stability increases exponentially with respect to the smoothness of the potential, in a sense to be made precise. The others show how the first estimate depends on the energy. In particular it is found that for high energies the stability estimate changes, in some sense, from logarithmic type to Lipschitz type: in this sense the ill-posedness of the problem decreases with increasing energy (in modulus).     

柔性约束下压杆的一些稳定和不稳定的临界状态

研究了一端固定、一端弹簧约束滑动固定的压杆在Euler临界载荷作用下的稳定性.将系统的势能表示为转角的泛函,将扰动量展开成Fourier级数,将势能的二阶变分表示成一个二次型,得到在临界状态下势能的二阶变分半正定,并求得临界载荷与屈曲模态.进一步研究临界状态下高阶变分的正定性,包括四阶和六阶变分的正定性.结果表明,与刚性约束不同的是,柔性约束压杆临界状态的稳定性与约束的刚度有关,有稳定与不稳定之分,并给出了临界状态是稳定和不稳定的情况下柔性约束相对刚度的范围.     

A note on strong duality in convex semidefinite optimization: necessary and sufficient conditions

A strong duality which states that the optimal values of the primal convex problem and its Lagrangian dual problem are equal (i.e. zero duality gap) and the dual problem attains its maximum is a corner stone in convex optimization. In particular it plays a major role in the numerical solution as well as the application of convex semidefinite optimization. The strong duality requires a technical condition known as a constraint qualification (CQ). Several CQs which are sufficient for strong duality have been given in the literature. In this note we present new necessary and sufficient CQs for the strong duality in convex semidefinite optimization. These CQs are shown to be sharper forms of the strong conical hull intersection property (CHIP) of the intersecting sets of constraints which has played a critical role in other areas of convex optimization such as constrained approximation and error bounds. Research was partially supported by the Australian Research Council. The author is grateful to the referees for their helpful comments     

Variational Problems with a p-Homogeneous Energy

We investigate existence, uniqueness and positivity of minimizers or critical points for an energy functional which contains only p-homogeneous and linear terms, 1p-homogeneous part of the energy functional is that it be given by the p-th power of an equivalent, uniformly convex norm on the underlying Sobolev space. Finally, continuous dependence of minimizers on the energy functional is established.     

EXISTENCE OF MINIMIZING SOLUTIONS AROUND ＂EXTENDED STATES＂ FOR A NONLINEARLY ELASTIC CLAMPED PLANE MEMBRANE

The formal asymptotic analysis of D. Fox, A. Raoult & J.C. Simo has justified the two-dimensional nonlinear “membrane“ equations for a plate made of a Saint Venant-Kirchhoffmaterial,     

EXISTENCE OF MINIMIZING SOLUTIONS AROUND ``EXTENDED STATES' FOR A NONLINEARLY ELASTIC CLAMPED PLANE MEMBRANE

The formal asymptotic analysis of D. Fox, A. Raoult $\&$ J.C. Simo$^{[10]}$ has justified the two-dimensional nonlinear ``membrane' equations for a plate made of a Saint Venant-Kirchhoff material. This model, which retains the material-frame indifference of the original three dimensional problem in the sense that its energy density is invariant under the rotations of $\R^3$, is equivalent to finding the critical points of a functional whose nonlinear part depends on the first fundamental form of the unknown deformed surface. The author establishes here, by the inverse function theorem, the existence of an injective solution to the clamped membrane problem around particular forces corresponding physically to an ``extension' of the membrane. Furthermore, it is proved that the solution found in this fashion is also the unique minimizer to the nonlinear membrane functional, which is not sequentially weakly lower semi-continuous.     

On the problem of optimizing contact force distributions

The problem of optimizing the distribution of contact forces between a rigid obstacle and a discretized linear elastic body is considered. The design variables are the initial gaps between the potential contact nodal points and the obstacle. Two different cost functionals are investigated: the first reflects the objective of minimizing the maximum contact force; the second is the equilibrium potential energy. Contrary to what has been claimed in the literature, it is shown that these cost functionals do not give, in general, the same optimal design. However, it is also shown that, if a certain frequently realized assumption is met by the system flexibility matrix, then this equality does hold.The min-max cost functional is nonconvex and nondifferentiable, and Clarke's theory of nonsmooth optimization is used to establish a sufficient optimality condition. Investigating its consequences, both necessary and sufficient optimality conditions can be given. The equilibrium potential energy cost functional, on the other hand, turns out to have the remarkable porperties of differentiability and convexity.This work was supported by The Center for Industrial Information Technology (CENHT), Linköping Institute of Technology, Linköping, Sweden.     

Primal-dual stability in continuous linear optimization

Any linear (ordinary or semi-infinite) optimization problem, and also its dual problem, can be classified as either inconsistent or bounded or unbounded, giving rise to nine duality states, three of them being precluded by the weak duality theorem. The remaining six duality states are possible in linear semi-infinite programming whereas two of them are precluded in linear programming as a consequence of the existence theorem and the non-homogeneous Farkas Lemma. This paper characterizes the linear programs and the continuous linear semi-infinite programs whose duality state is preserved by sufficiently small perturbations of all the data. Moreover, it shows that almost all linear programs satisfy this stability property.     

Bifurcation of nontrivial periodic solutions for a biochemical model with impulsive perturbations

In this paper, a biochemical model with the impulsive perturbations is considered. By using the Floquet theorem, we find the boundary-periodic solution is asymptotically stable if the impulsive period is larger than a critical value. On the contrary, it is unstable if the impulsive period is less than the critical value. The problem of finding nontrivial periodic solutions is reduced to showing the existence of the nontrivial fixed points for the associated stroboscopic mapping of time snapshot equal to the common period of input. It is then shown that once a threshold condition is reached, a stable nontrivial periodic solution emerges via a supercritical bifurcation. Furthermore, influences of the impulsive input on the inherent oscillations are studied numerically, which shows the rich dynamics in the positive octant.