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C. H. Jeffrey Pang

Publications and source records attributed to C. H. Jeffrey Pang.

29 records · Page 2Linked to original sources
arXiv2012

Set intersection problems: Supporting hyperplanes and quadratic programming

We study how the supporting hyperplanes produced by the projection process can complement the method of alternating projections and its variants for the convex set intersection problem. For the problem of finding the closest point in the intersection of closed convex sets, we propose an algorithm that, like Dykstra's algorithm, converges strongly in a Hilbert space. Moreover, this algorithm converges in finitely many iterations when the closed convex sets are cones in $\mathbb{R}^{n}$ satisfying an alignment condition. Next, we propose modifications of the alternating projection algorithm, and prove its convergence. The algorithm converges superlinearly in $\mathbb{R}^{n}$ under some nice conditions. Under a conical condition, the convergence can be finite. Lastly, we discuss the case where the intersection of the sets is empty.

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arXiv2012

Characterizing generalized derivatives of set-valued maps: Extending the tangential and normal approaches

For a set-valued map, we characterize, in terms of its (unconvexified or convexified) graphical derivatives near the point of interest, positively homogeneous maps that are generalized derivatives in the sense of [20]. This result generalizes the Aubin criterion in [9]. A second characterization of these generalized derivatives is easier to check in practice, especially in the finite dimensional case. Finally, the third characterization in terms of limiting normal cones and coderivatives generalizes the Mordukhovich criterion in the finite dimensional case. The convexified coderivative has a bijective relationship with the set of possible generalized derivatives. We conclude by illustrating a few applications of our result.

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arXiv2012

Finding saddle points of mountain pass type with quadratic models on affine spaces

The problem of computing saddle points is important in certain problems in numerical partial differential equations and computational chemistry, and is often solved numerically by a minimization problem over a set of mountain passes. We propose an algorithm to find saddle points of mountain pass type to find the bottlenecks of optimal mountain passes. The key step is to minimize the distance between level sets by using quadratic models on affine spaces similar to the strategy in the conjugate gradient algorithm. We discuss parameter choices, convergence results, and how to augment the algorithm to a path based method. Finally, we perform numerical experiments to test the convergence of our algorithm.

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arXiv2012

Some principles for mountain pass algorithms, and the parallel distance

The problem of computing saddle points is important in certain problems in numerical partial differential equations and computational chemistry, and is often solved numerically by a minimization problem over a set of mountain passes. We point out that a good global mountain pass algorithm should have good local and global properties. Next, we define the parallel distance, and show that the square of the parallel distance has a quadratic property. We show how to design algorithms for the mountain pass problem based on perturbing parameters of the parallel distance, and that methods based on the parallel distance have midrange local and global properties.

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arXiv2012

Subdifferential analysis of differential inclusions via discretization

The framework of differential inclusions encompasses modern optimal control and the calculus of variations. Necessary optimality conditions in the literature identify potentially optimal paths, but do not show how to perturb paths to optimality. We first look at the corresponding discretized inclusions, estimating the subdifferential dependence of the optimal value in terms of the endpoints of the feasible paths. Our approach is to first estimate the coderivative of the reachable map. The discretized (nonsmooth) Euler-Lagrange and transversality conditions follow as a corollary. We obtain corresponding results for differential inclusions by passing discretized inclusions to the limit.

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arXiv2010

Generalized differentiation with positively homogeneous maps: Applications in set-valued analysis and metric regularity

We propose a new concept of generalized differentiation of set-valued maps that captures the first order information. This concept encompasses the standard notions of Frechet differentiability, strict differentiability, calmness and Lipschitz continuity in single-valued maps, and the Aubin property and Lipschitz continuity in set-valued maps. We present calculus rules, sharpen the relationship between the Aubin property and coderivatives, and study how metric regularity and open covering can be refined to have a directional property similar to our concept of generalized differentiation. Finally, we discuss the relationship between the robust form of generalization differentiation and its one sided counterpart.

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arXiv2010

Level set methods for finding critical points of mountain pass type

Computing mountain passes is a standard way of finding critical points. We describe a numerical method for finding critical points that is convergent in the nonsmooth case and locally superlinearly convergent in the smooth finite dimensional case. We apply these techniques to describe a strategy for the Wilkinson problem of calculating the distance of a matrix to a closest matrix with repeated eigenvalues. Finally, we relate critical points of mountain pass type to nonsmooth and metric critical point theory.

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arXiv2010

Level set methods for finding saddle points of general Morse index

For a real valued function, a point is critical if its derivatives are zero, and a critical point is a saddle point if it is not a local extrema. In this paper, we study algorithms to find saddle points of general Morse index. Our approach is motivated by the multidimensional mountain pass theorem, and extends our earlier work on methods (based on studying the level sets) to find saddle points of mountain pass type. We prove the convergence of our algorithms in the nonsmooth case, and the local superlinear convergence of another algorithm in the smooth finite dimensional case.

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arXiv2009

Continuity of set-valued maps revisited in the light of tame geometry

Continuity of set-valued maps is hereby revisited: after recalling some basic concepts of variational analysis and a short description of the State-of-the-Art, we obtain as by-product two Sard type results concerning local minima of scalar and vector valued functions. Our main result though, is inscribed in the framework of tame geometry, stating that a closed-valued semialgebraic set-valued map is almost everywhere continuous (in both topological and measure-theoretic sense). The result, depending on stratification techniques, holds true in a more general setting of o-minimal (or tame) set-valued maps. Some applications are briefly discussed at the end.

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arXiv2008

Lipschitz behavior of the robust regularization

To minimize or upper-bound the value of a function "robustly", we might instead minimize or upper-bound the "epsilon-robust regularization", defined as the map from a point to the maximum value of the function within an epsilon-radius. This regularization may be easy to compute: convex quadratics lead to semidefinite-representable regularizations, for example, and the spectral radius of a matrix leads to pseudospectral computations. For favorable classes of functions, we show that the robust regularization is Lipschitz around any given point, for all small epsilon > 0, even if the original function is nonlipschitz (like the spectral radius). One such favorable class consists of the semi-algebraic functions. Such functions have graphs that are finite unions of sets defined by finitely-many polynomial inequalities, and are commonly encountered in applications.

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