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We construct three new combinatorial Hopf algebras based on the Loday-Ronco operations on planar binary trees. The first and second algebras are defined on planar trees and labeled planar trees extending the Loday-Ronco and Malvenuto-Reutenauer Hopf algebras respectively. We show that the latter is bidendriform which implies that is also free, cofree, and self-dual. The third algebra involves a new visualization of parking functions as decorated binary trees; it is also bidendriform, free, cofree, and self-dual, and therefore abstractly isomorphic to the algebra PQSym of Novelli and Thibon. We define partial orders on the objects indexing each of these three Hopf algebras, one of which, when restricting to (m+1)-ary trees, coarsens the m-Tamari order of Bergeron and Pr\'eville-Ratelle. We show that multiplication of dual fundamental basis elements are given by intervals in each of these orders. Finally, we use an axiomatized version of the techniques of Aguiar and Sottile on the Malvenuto-Reutenauer Hopf algebra to define a monomial basis on each of our Hopf algebras, and to show that comultiplication is cofree on the monomial elements. This in particular, implies the cofreeness of the Hopf algebra on planar trees. We also find explicit positive formulas for the multiplication on monomial basis and a cancellation-free and grouping-free formula for the antipode of monomial elements.

We extend an algebra of Mantaci and Reutenauer, acting on the free associative algebra, to a vector space of operators acting on all graded connected Hopf algebras. These operators are convolution products of certain involutions, which we view as hyperoctahedral variants of Patras's descent operators. We obtain the eigenvalues and multiplicities of all our new operators, as well as a basis of eigenvectors for a subclass akin to Adams operations. We outline how to apply this eigendata to study Markov chains, and examine in detail the case of card-shuffles with flips or rotations., Comment: 40 pages, somewhat due to many multi-line equations. Comments are very welcome!

We show the linear convergence of Dykstra's algorithm for sets intersecting in a manner slightly stronger than the usual constraint qualifications., Comment: Some inaccuracies in the treatment of case 3b is corrected. More elaboration in other parts to make reading of the proof easier

This work builds on our recent work on a distributed optimization algorithm for graphs with directed unreliable communications. We show its linear convergence when we take either the proximal of each function or an affine minorant for when the function is smooth., Comment: 18 pages, 2 figures

We investigate the convergence rate of the distributed Dykstra's algorithm when some of the sets are defined as the level sets of convex functions. We carry out numerical experiments to compare with the theoretical results obtained., Comment: 18 pages, 4 figures

We show that the averaged consensus algorithm on directed graphs with unreliable communications by Bof-Carli-Schenato has a dual optimization interpretation, which could be extended to the case of distributed optimization. We report on our numerical simulations for the distributed optimization algorithm for smooth and nonsmooth functions., Comment: 13 pages, 2 figures. Elaborated proofs, mostly in Theorems 3.2 and 3.3. Added a figure to describe the algorithm of Bof-Carli-Schenato

In two earlier papers, we designed a distributed deterministic asynchronous algorithm for minimizing the sum of subdifferentiable and proximable functions and a regularizing quadratic on time-varying graphs based on Dykstra's algorithm, or block coordinate dual ascent. Each node in the distributed optimization problem is the sum of a known regularizing quadratic and a function to be minimized. In this paper, we prove sublinear convergence rates for the general algorithm, and a linear rate of convergence if the function on each node is smooth with Lipschitz gradient., Comment: 29 pages. New in this submission: Numerical experiments, and some updates to improve clarity

We described a decentralized distributed deterministic asynchronous Dykstra's algorithm that allows for time-varying graphs in an earlier paper. In this paper, we show how to incorporate subdifferentiable functions into the framework using a step similar to the bundle method. We point out that our algorithm also allows for partial data communications. We discuss a standard step for treating the composition of a convex and linear function., Comment: 29 pages. New in this submission: Subsection 1.3 describes the contributions of this paper without assuming knowledge from other papers

Consider the setting where each vertex of a graph has a function, and communications can only occur between vertices connected by an edge. We wish to minimize the sum of these functions. For the case when each function is the sum of a strongly convex quadratic and a convex function, we propose a distributed version of Dykstra's algorithm. The computations to optimize the dual objective function can run asynchronously without a global clock, and in a distributed manner without a central controller. Convergence to the primal minimizer is deterministic instead of being probabilistic, and is guaranteed as long as in each cycle, the edges where two-way communications occur connects all vertices. We also look at an accelerated algorithm, and an algorithm for the case when the functions on the nodes are not strongly convex., Comment: 27 pages. Accepted in SIAM J. Optim. . Added Remark 2.6. Added references on the dual ascent perspective, and made other small changes. We thank the anonymous referees, associate editor and the journal staff for the quick processing of my submission

We show a general framework of parallelizing Dykstra splitting that includes the classical Dykstra's algorithm and the product space formulation as special cases, and prove their convergence. The key idea is to split up the function whose conjugate takes in the sum of all dual variables in the dual formulation., Comment: 15 pages

We show that Dykstra's splitting for projecting onto the intersection of convex sets can be extended to minimize the sum of convex functions and a regularizing quadratic. We give conditions for which convergence to the primal minimizer holds so that more than one convex function can be minimized at a time, the convex functions are not necessarily sampled in a cyclic manner, and the SHQP strategy for problems involving the intersection of more than one convex set can be applied. When the sum does not involve the regularizing quadratic, we discuss an approximate proximal point method combined with Dykstra's splitting to minimize this sum., Comment: 24 pages

We show that an almost cyclic (or generalized Gauss- Seidel) Dykstra's algorithm which incorporates the SHQP (supporting halfspace- quadratic programming) strategy can achieve nonasymptotic and asymptotic linear convergence for polyhedral problems., Comment: 26 pages

The interaction of water with TiO2 is crucial to many of its practical applications, including photocatalytic water splitting. Following the first demonstration of this phenomenon 40 years ago there have been numerous studies of the rutile single-crystal TiO2(110) interface with water. This has provided an atomic-level understanding of the water-TiO2 interaction. However, nearly all of the previous studies of water/TiO2 interfaces involve water in the vapour phase. Here, we explore the interfacial structure between liquid water and a rutile TiO2(110) surface pre-characterized at the atomic level. Scanning tunnelling microscopy and surface X-ray diffraction are used to determine the structure, which is comprised of an ordered array of hydroxyl molecules with molecular water in the second layer. Static and dynamic density functional theory calculations suggest that a possible mechanism for formation of the hydroxyl overlayer involves the mixed adsorption of O2 and H2O on a partially defected surface. The quantitative structural properties derived here provide a basis with which to explore the atomistic properties and hence mechanisms involved in TiO2 photocatalysis.

We develop a general theory for Markov chains whose transition probabilities are the coefficients of descent operators on combinatorial Hopf algebras. These model the breaking-then-recombining of combinational objects. Examples include the various card-shuffles of Diaconis, Fill and Pitman, Fulman's restriction-then-induction chains on the representations of the symmetric group, and a plethora of new chains on trees, partitions and permutations. The eigenvalues of these chains can be calculated in a uniform manner using Hopf algebra structure theory, and there is a simple expression for their stationary distributions. For an important subclass of chains analogous to the top-to-random shuffle, we derive a full right eigenbasis, from which follow exact expressions for expectations of certain statistics of interest. This greatly generalises the coproduct-then-product chains previously studied in joint work with Persi Diaconis and Arun Ram., Comment: 57 pages. Sections 4.4, 5.3 and 6 are new. Sections 5 and 6 may swap order in a future version, so be sure to cite the version number!

The number of alternating runs is a natural permutation statistic. We show it can be used to define some commutative subalgebras of the symmetric group algebra, and more precisely of the descent algebra. The Eulerian peak algebras naturally appear as subalgebras of our run algebras. We also calculate the orthogonal idempotents for run algebras in terms of noncommutative symmetric functions.

We consider the best approximation problem (BAP) of projecting a point onto the intersection of a number of convex sets. It is known that Dykstra's algorithm is alternating minimization on the dual problem. We extend Dykstra's algorithm so that it can be enhanced by the SHQP strategy of using quadratic programming to project onto the intersection of supporting halfspaces generated by earlier projection operations. By looking at a structured alternating minimization problem, we show the convergence rate of Dykstra's algorithm when reasonable conditions are imposed to guarantee a dual minimizer. We also establish convergence of using a warmstart iterate for Dykstra's algorithm, show how all the results for the Dykstra's algorithm can be carried over to the simultaneous Dykstra's algorithm, and discuss a different way of incorporating the SHQP strategy. Lastly, we show that the dual of the best approximation problem can have an O(1/k^2) accelerated algorithm that also incorporates the SHQP strategy., Comment: 26 pages, 1 figure

A function on the state space of a Markov chain is a "lumping" if observing only the function values gives a Markov chain. We give very general conditions for lumpings of a large class of algebraically-defined Markov chains, which include random walks on groups and other common constructions. We specialise these criteria to the case of descent operator chains from combinatorial Hopf algebras, and, as an example, construct a "top-to-random-with-standardisation" chain on permutations that lumps to a popular restriction-then-induction chain on partitions, using the fact that the algebra of symmetric functions is a subquotient of the Malvenuto-Reutenauer algebra., Comment: 36 pages. Minor update from v2 after peer-review, with numerical sectioning as in the published version. (v2 was majorly reworked from v1, with new results in sections 3.5/II.D and 3.6/II.E, and new examples in Part I/Section 2.)

The problem of finding a point in the intersection of closed sets can be solved by the method of alternating projections and its variants. It was shown in earlier papers that for convex sets, the strategy of using quadratic programming (QP) to project onto the intersection of supporting halfspaces generated earlier by the projection process can lead to an algorithm that converges multiple-term superlinearly. The main contributions of this paper are to show that this strategy can be effective for super-regular sets, which are structured nonconvex sets introduced by Lewis, Luke and Malick. Manifolds should be approximated by hyperplanes rather than halfspaces. We prove the linear convergence of this strategy, followed by proving that superlinear and quadratic convergence can be obtained when the problem is similar to the setting of the Newton method. We also show an algorithm that converges at an arbitrarily fast linear rate if halfspaces from older iterations are used to construct the QP., Comment: 25 pages, 4 figures

We propose first order algorithms for convex optimization problems where the feasible set is described by a large number of convex inequalities that is to be explored by subgradient projections. The first algorithm is an adaptation of a subgradient algorithm, and has convergence rate $1/\sqrt{k}$. The second algorithm has convergence rate 1/k when (1) one has linear metric inequality in the feasible set, (2) the objective function is strongly convex, differentiable and has Lipschitz gradient, and (3) it is easy to optimize the objective function on the intersection of two halfspaces. This second algorithm generalizes Haugazeau's algorithm. The third algorithm adapts the second algorithm when condition (3) is dropped. We give examples to show that the second algorithm performs poorly when the objective function is not strongly convex, or when the linear metric inequality is absent., Comment: 26 pages, 2 figures

Recently, Diaconis, Ram and I created Markov chains out of the coproduct-then-product operator on combinatorial Hopf algebras. These chains model the breaking and recombining of combinatorial objects. Our motivating example was the riffle-shuffling of a deck of cards, for which this Hopf algebra connection allowed explicit computation of all the eigenfunctions. The present note replaces in this construction the coproduct-then-product map with convolutions of projections to the graded subspaces, effectively allowing us to dictate the distribution of sizes of the pieces in the breaking step of the previous chains. An important example is removing one "vertex" and reattaching it, in analogy with top-to-random shuffling. This larger family of Markov chains all admit analysis by Hopf-algebraic techniques. There are simple combinatorial expressions for their stationary distributions and for their eigenvalues and multiplicities and, in some cases, the eigenfunctions are also calculable., Comment: 12 pages. This is an extended abstract, to appear in Proceedings of the 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC). Comments are very welcome

This thesis introduces a way to build Markov chains out of Hopf algebras. The transition matrix of a "Hopf-power Markov chain" is (the transpose of) the matrix of the coproduct-then-product operator on a combinatorial Hopf algebra with respect to a suitable basis. These chains describe the breaking-then-recombining of the combinatorial objects in the Hopf algebra. The motivating example is the famous Gilbert-Shannon-Reeds model of riffle-shuffling of a deck of cards, which arises in this manner from the shuffle algebra. The primary reason for constructing Hopf-power Markov chains, or for rephrasing familiar chains through this lens, is that much information about them comes simply from translating well-known facts on the underlying Hopf algebra. For example, there is an explicit formula for the stationary distribution (Theorem 4.5.1), and constructing quotient algebras show that certain statistics on a Hopf-power Markov chain are themselves Markov chains (Theorem 4.7.1). Perhaps the pinnacle is Theorem 2.5.1, a collection of algorithms for a full left and right eigenbasis in many common cases where the underlying Hopf algebra is commutative or cocommutative. This arises from a cocktail of the Poincare-Birkhoff-Witt theorem, the Cartier-Milnor-Moore theorem, Reutenauer's structure theory of the free Lie algebra, and Patras's Eulerian idempotent theory. Since Hopf-power Markov chains can exhibit very different behaviour depending on the structure of the underlying Hopf algebra and its distinguished basis, one must restrict attention to certain styles of Hopf algebras in order to obtain stronger results. This thesis will focus respectively on a free-commutative basis, which produces "independent breaking" chains, and a cofree basis; there will be both general statements and in-depth examples., Comment: Revised version of PhD thesis. 178 pages, 6 figures. Comments are very welcome. For updated version, see my webpage. In v2: fixed references in table at end of Chapter 1

A known first order method to find a feasible solution to a conic problem is an adapted von Neumann algorithm. We improve the distance reduction step there by projecting onto the convex hull of previously generated points using a primal active set quadratic programming (QP) algorithm. The convergence theory is improved when the QPs are as large as possible. For problems in R^2, we analyze our algorithm by epigraphs and the monotonicity of subdifferentials. Logically, the larger the set to project onto, the better the performance per iteration, and this is indeed seen in our numerical experiments., Comment: 28 pages, 9 figures

This paper improves the algorithms based on supporting halfspaces and quadratic programming for convex set intersection problems in our earlier paper in several directions. First, we give conditions so that much smaller quadratic programs (QPs) and approximate projections arising from partially solving the QPs are sufficient for multiple-term superlinear convergence for nonsmooth problems. Second, we identify additional regularity, which we call the second order supporting hyperplane property (SOSH), that gives multiple-term quadratic convergence. Third, we show that these fast convergence results carry over for the convex inequality problem. Fourth, we show that infeasibility can be detected in finitely many operations. Lastly, we explain how we can use the dual active set QP algorithm of Goldfarb and Idnani to get useful iterates by solving the QPs partially, overcoming the problem of solving large QPs in our algorithms., Comment: 27 pages, 2 figures

The von Neumann-Halperin method of alternating projections converges strongly to the projection of a given point onto the intersection of finitely many closed affine subspaces. We propose acceleration schemes making use of two ideas: Firstly, each projection onto an affine subspace identifies a hyperplane of codimension 1 containing the intersection, and secondly, it is easy to project onto a finite intersection of such hyperplanes. We give conditions for which our accelerations converge strongly. Finally, we perform numerical experiments to show that these accelerations perform well for a matrix model updating problem., Comment: 16 pages, 3 figures (Corrected minor typos in Remark 2.2, Algorithm 2.5, proof of Theorem 3.12, as well as elaborated on certain proofs

We extend Fukushima's result on the finite convergence of an algorithm for the global convex feasibility problem to the local nonconvex case., Comment: Added a figure to explain an inequality. Added elaboration of sublinear. Corrected some typos

Abstract. The Set Intersection Problem (SIP) is the problem of finding a point in the intersection of convex sets. This problem is typically solved by the method of alternating projections. To accelerate the convergence, the idea of using Quadratic Programming (QP) to project a point onto the intersection of halfspaces generated by the projection process was discussed in earlier papers. This paper looks at how one can integrate projection algorithms together with an active set QP algorithm. As a byproduct of our analysis, we show how to accelerate an SIP algorithm involving box constraints, and how to extend a version of the Algebraic Reconstruction Technique (ART) while preserving finite convergence. Lastly, the warmstart property of active set QP algorithms is a valuable property for the problem of projecting onto the intersection of convex sets., Comment: 25 pages, 7 figures. This submission is completely different from the last submission. I now feel that the last submission didn't develop the idea of integrating the dual active set quadratic programming algorithm of Goldfarb and Idnani to projection algorithms well enough, and that this submission has developed the idea and worked out relevant theoretical issues much better

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., Comment: 25 pages, 3 figures

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., Comment: 24 Pages, 5 Figures

The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural "rock-breaking" process, and Markov chains on simplicial complexes. Many of these chains can be explictly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rock-breaking, an explicit description of the quasi-stationary distribution and sharp rates to absorption follow., Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes will only appear on the version on Amy Pang's website, the arXiv version will not be updated.)

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., Comment: I just want to change the comments portion (otherwise article is same as v2). Changes in published version: The assumption that F is osc is added in a few places. The condition (1) in Lemma 5.9 is improved slightly. Added further references, commentary and acknowledgments

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., Comment: This paper has been withdrawn because the important ideas have been poorly developed here. A new submission has since subsumed the material in this paper. 20 pages, 4 figures

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., Comment: Accepted for publication at SIAM. J. Control Optim.. This submission is not the final version, but corrects the first version in various places

We show that a first order problem can approximate solutions of a robust optimization problem when the uncertainty set is scaled, and explore further properties of this first order problem., Comment: 25 pages, 3 figures