Showing papers on "Monotone polygon published in 2016"

Models of dielectric relaxation based on completely monotone functions

Abstract: Abstract The relaxation properties of dielectric materials are described, in the frequency domain, according to one of the several models proposed over the years: Kohlrausch-Williams-Watts, Cole-Cole, Cole-Davidson, Havriliak-Negami (with its modified version) and Excess wing model are among the most famous. Their description in the time domain involves some mathematical functions whose knowledge is of fundamental importance for a full understanding of the models. In this work, we survey the main dielectric models and we illustrate the corresponding time-domain functions. In particular, we stress the attention on the completely monotone character of the relaxation and response functions. We also provide a characterization of the models in terms of differential operators of fractional order.

An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems

Abstract: We introduce and investigate the convergence properties of an inertial forward-backward-forward splitting algorithm for approaching the set of zeros of the sum of a maximally monotone operator and a single-valued monotone and Lipschitzian operator. By making use of the product space approach, we expand it to the solving of inclusion problems involving mixtures of linearly composed and parallel-sum type monotone operators. We obtain in this way an inertial forward-backward-forward primal-dual splitting algorithm having as main characteristic the fact that in the iterative scheme all operators are accessed separately either via forward or via backward evaluations. We present also the variational case when one is interested in the solving of a primal-dual pair of convex optimization problems with complexly structured objectives, which we also illustrate by numerical experiments in image processing.

Guaranteed Non-convex Optimization: Submodular Maximization over Continuous Domains

Abstract: Submodular continuous functions are a category of (generally) non-convex/non-concave functions with a wide spectrum of applications. We characterize these functions and demonstrate that they can be maximized efficiently with approximation guarantees. Specifically, i) We introduce the weak DR property that gives a unified characterization of submodularity for all set, integer-lattice and continuous functions; ii) for maximizing monotone DR-submodular continuous functions under general down-closed convex constraints, we propose a Frank-Wolfe variant with $(1-1/e)$ approximation guarantee, and sub-linear convergence rate; iii) for maximizing general non-monotone submodular continuous functions subject to box constraints, we propose a DoubleGreedy algorithm with $1/3$ approximation guarantee. Submodular continuous functions naturally find applications in various real-world settings, including influence and revenue maximization with continuous assignments, sensor energy management, multi-resolution data summarization, facility location, etc. Experimental results show that the proposed algorithms efficiently generate superior solutions compared to baseline algorithms.

Stochastic Variance Reduction Methods for Saddle-Point Problems

Abstract: We consider convex-concave saddle-point problems where the objective functions may be split in many components, and extend recent stochastic variance reduction methods (such as SVRG or SAGA) to provide the first large-scale linearly convergent algorithms for this class of problems which are common in machine learning. While the algorithmic extension is straightforward, it comes with challenges and opportunities: (a) the convex minimization analysis does not apply and we use the notion of monotone operators to prove convergence, showing in particular that the same algorithm applies to a larger class of problems, such as variational inequalities, (b) there are two notions of splits, in terms of functions, or in terms of partial derivatives, (c) the split does need to be done with convex-concave terms, (d) non-uniform sampling is key to an efficient algorithm, both in theory and practice, and (e) these incremental algorithms can be easily accelerated using a simple extension of the "catalyst" framework, leading to an algorithm which is always superior to accelerated batch algorithms.

BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration

Abstract: We analyze multidimensional BSDEs in a filtration that supports a Brownian motion and a Poisson random measure. Under a monotonicity assumption on the driver, the paper extends several results from the literature. We establish existence and uniqueness of solutions in provided that the generator and the terminal condition satisfy appropriate integrability conditions. The analysis is first carried out under a deterministic time horizon, and then generalized to random time horizons given by a stopping time with respect to the underlying filtration. Moreover, we provide a comparison principle in dimension one.

The split common null point problem and the shrinking projection method in Banach spaces

Abstract: In this paper, we consider the split common null point problem with resolvents of maximal monotone operators in Banach spaces. Then, using the shrinking projection method, we prove a strong convergence theorem for finding a solution of the split common null point problem in Banach spaces.

Maximizing Stochastic Monotone Submodular Functions

Abstract: We study the problem of maximizing a stochastic monotone submodular function with respect to a matroid constraint. Because of the presence of diminishing marginal values in real-world problems, our model can capture the effect of stochasticity in a wide range of applications. We show that the adaptivity gap—the ratio between the values of optimal adaptive and optimal nonadaptive policies—is bounded and is equal to e/(e − 1). We propose a polynomial-time nonadaptive policy that achieves this bound. We also present an adaptive myopic policy that obtains at least half of the optimal value. Furthermore, when the matroid is uniform, the myopic policy achieves the optimal approximation ratio of 1 − 1/e. This paper was accepted by Dimitris Bertsimas and Yinyu Ye, optimization.

Efficiently solving general rearrangement tasks: A fast extension primitive for an incremental sampling-based planner

Abstract: Manipulating multiple movable obstacles is a hard problem that involves searching high-dimensional C-spaces. A milestone method for this problem was able to compute solutions for monotone instances. These are problems where every object needs to be transferred at most once to achieve a desired arrangement. The method uses backtracking search to find the order with which objects should be moved. This paper first proposes an approximate but significantly faster alternative for monotone rearrangement instances. The method defines a dependency graph between objects given minimum constraint removal paths (MCR) to transfer each object to its target. From this graph, the approach discovers the order of moving objects by performing topological sorting without backtracking search. The approximation arises from the limitation to consider only MCR paths, which minimize, however, the number of conflicts between objects. To solve non-monotone instances, this primitive is incorporated in a higher-level incremental search algorithm for general rearrangement planning, which operates similar to Bi-RRT. Given a start and a goal object arrangement, tree structures of reachable new arrangements are generated by using the primitive as an expansion procedure. The integrated solution achieves probabilistic completeness for the general non-monotone case and based on simulated experiments it achieves very good success ratios, solution times and path quality relative to alternatives.

Convergence of One-Step Projected Gradient Methods for Variational Inequalities

Abstract: In this paper, we revisit the numerical approach to some classical variational inequalities, with monotone and Lipschitz continuous mapping A, by means of a projected reflected gradient-type method. A main feature of the method is that it formally requires only one projection step onto the feasible set and one evaluation of the involved mapping per iteration. Contrary to what was done so far, we establish the convergence of the method in a more general setting that allows us to use varying step-sizes without any requirement of additional projections. A linear convergence rate is obtained, when A is assumed to be strongly monotone. Preliminary numerical experiments are also performed.

Exponential Lower Bounds for Monotone Span Programs

Abstract: Monotone span programs are a linear-algebraic model of computation which were introduced by Karchmer and Wigderson in 1993 [1]. They are known to be equivalent to linear secret sharing schemes, and have various applications in complexity theory and cryptography. Lower bounds for monotone span programs have been difficult to obtain because they use non-monotone operations to compute monotone functions, in fact, the best known lower bounds are quasipolynomial for a function in (nonmonotone) P [2]. A fundamental open problem is to prove exponential lower bounds on monotone span program size for any explicit function. We resolve this open problem by giving exponential lower bounds on monotone span program size for a function in monotone P. This also implies the first exponential lower bounds for linear secret sharing schemes. Our result is obtained by proving exponential lower bounds using Razborov's rank method [3], a measure that is strong enough to prove lower bounds for many monotone models. As corollaries we obtain new proofs of exponential lower bounds for monotone formula size, monotone switching network size, and the first lower bounds for monotone comparator circuit size for a function in monotone P. We also obtain new polynomial degree lower bounds for Nullstellensatz refutations using an interpolation theorem of Pudlak and Sgall [4]. Finally, we obtain quasipolynomial lower bounds on the rank measure for the st-connectivity function, implying tight bounds for st-connectivity in all of the computational models mentioned above.

Differentially Positive Systems

Abstract: The paper introduces and studies differentially positive systems, that is, systems whose linearization along an arbitrary trajectory is positive. A generalization of Perron–Frobenius theory is developed in this differential framework to show that the property induces a conal order that strongly constrains the asymptotic behavior of solutions. The results illustrate that behaviors constrained by local order properties extend beyond the well-studied class of linear positive systems and monotone systems, which both require a constant cone field and a linear state space.

New Approach to the Stability of Chemical Reaction Networks: Piecewise Linear in Rates Lyapunov Functions

Abstract: Piecewise-linear in rates (PWLR) Lyapunov functions are introduced for a class of chemical reaction networks (CRNs). In addition to their simple structure, these functions are robust with respect to arbitrary monotone reaction rates, of which Mass-Action is a special case. The existence of such functions ensures the convergence of trajectories towards equilibria, and can be used to establish their asymptotic stability with respect to the corresponding stoichiometric compatibility class. We give the definition of these Lyapunov functions, prove their basic properties, and provide algorithms for constructing them. Examples are provided, relationship with consensus dynamics are discussed, and future directions are elaborated.

Models of dielectric relaxation based on completely monotone functions

Abstract: The relaxation properties of dielectric materials are described, in the frequency domain, according to one of the several models proposed over the years: Kohlrausch-Williams-Watts, Cole-Cole, Cole-Davidson, Havriliak-Negami (with its modified version) and Excess wing model are among the most famous. Their description in the time domain involves some mathematical functions whose knowledge is of fundamental importance for a full understanding of the models. In this work, we survey the main dielectric models and we illustrate the corresponding time-domain functions. In particular, we stress the attention on the completely monotone character of the relaxation and response functions. We also provide a characterization of the models in terms of differential operators of fractional order.

Proximal methods for stationary Mean Field Games with local couplings

Abstract: We address the numerical approximation of Mean Field Games with local couplings. For power-like Hamiltonians, we consider both unconstrained and constrained stationary systems with density constraints in order to model hard congestion effects. For finite difference discretizations of the Mean Field Game system, we follow a variational approach. We prove that the aforementioned schemes can be obtained as the optimality system of suitably defined optimization problems. In order to prove the existence of solutions of the scheme with a variational argument, the monotonicity of the coupling term is not used, which allow us to recover general existence results. Next, assuming next that the coupling term is monotone, the variational problem is cast as a convex optimization problem for which we study and compare several proximal type methods. These algorithms have several interesting features, such as global convergence and stability with respect to the viscosity parameter, which can eventually be zero. We assess the performance of the methods via numerical experiments.

Monotone and consistent discretization of the Monge-Ampère operator

Abstract: We introduce a novel discretization of the Monge-Ampere operator, simultaneously consistent and degenerate elliptic, hence accurate and robust in applications. These properties are achieved by exploiting the arithmetic structure of the discrete domain, assumed to be a two dimensional cartesian grid. The construction of our scheme is simple, but its analysis relies on original tools seldom encountered in numerical analysis, such as the geometry of two dimensional lattices, and an arithmetic structure called the Stern-Brocot tree. Numerical experiments illustrate the method's efficiency.

Construction algorithms for a class of monotone variational inequalities

Abstract: This paper is devoted to solve the following monotone variational inequality of finding \(x^*\in \mathrm{Fix}(T)\) such that $$\begin{aligned} \langle Ax^*,x-x^*\rangle \ge 0,\quad \forall x\in \mathrm{Fix}(T), \end{aligned}$$ where A is a monotone operator and \(\mathrm{Fix}(T)\) is the set of fixed points of nonexpansive operator T. For this purpose, we construct an implicit algorithm and prove its convergence hierarchical to the solution of above monotone variational inequality.

Infinitely many exotic monotone Lagrangian tori in ℂℙ2

Abstract: In [10], we construct an exotic monotone Lagrangian torus in CP 2 (not Hamiltonian isotopic to the known Clifford and Chekanov tori) using techniques motivated by mirror symmetry. We named it T(1,4,25) because, when following a degeneration of CP 2 to the orbifold CP(1,4,25), it degenerates to the central fiber of the moment map for the standard torus action on CP(1,4,25). Related to each degeneration from CP 2 to CP(a 2 ,b 2 ,c 2 ), for (a,b,c) a Markov triple - see (1.1) - there is a monotone Lagrangian torus, which we call T(a 2 ,b 2 ,c 2 ). We conjectured that no two of them are Hamiltonian isotopic to each other. Here we employ techniques from symplectic field theory to prove that the above conjecture is true.

Linear Inviscid Damping for Monotone Shear Flows in a Finite Periodic Channel, Boundary Effects, Blow-up and Critical Sobolev Regularity

Abstract: In a previous article (Zillinger, Linear inviscid damping for monotone shear flows, 2014), we have established linear inviscid damping for a large class of monotone shear flows in a finite periodic channel and have further shown that boundary effects asymptotically lead to the formation of singularities of derivatives of the solution as \({t \rightarrow \infty}\). As the main results of this article, we provide a detailed description of the singularity formation and establish stability in all sub-critical fractional Sobolev spaces and blow-up in all super-critical spaces. Furthermore, we discuss the implications of the blow-up to the problem of nonlinear inviscid damping in a finite periodic channel, where high regularity would be essential to control nonlinear effects.

A note on the triangle inequality for the Jaccard distance

Abstract: Two simple proofs of the triangle inequality for the Jaccard distance in terms of nonnegative, monotone, submodular functions are given and discussed.

Ramifications of Hurwitz theory, KP integrability and quantum curves

Abstract: In this paper we revisit several recent results on monotone and strictly monotone Hurwitz numbers, providing new proofs. In particular, we use various versions of these numbers to discuss methods of derivation of quantum spectral curves from the point of view of KP integrability and derive new examples of quantum curves for the families of double Hurwitz numbers.

Stochastic Forward---Backward Splitting for Monotone Inclusions

Abstract: We propose and analyze the convergence of a novel stochastic algorithm for monotone inclusions that are sum of a maximal monotone operator and a single-valued cocoercive operator. The algorithm we propose is a natural stochastic extension of the classical forward---backward method. We provide a non-asymptotic error analysis in expectation for the strongly monotone case, as well as almost sure convergence under weaker assumptions. For minimization problems, we recover rates matching those obtained by stochastic extensions of the so-called accelerated methods. Stochastic quasi-Fejer's sequences are a key technical tool to prove almost sure convergence.

Stability of Traveling Pulses with Oscillatory Tails in the FitzHugh–Nagumo System

Abstract: The FitzHugh–Nagumo equations are known to admit fast traveling pulses that have monotone tails and arise as the concatenation of Nagumo fronts and backs in an appropriate singular limit, where a parameter $$\varepsilon $$ goes to zero. These pulses are known to be nonlinearly stable with respect to the underlying PDE. Recently, the existence of fast pulses with oscillatory tails was proved for the FitzHugh–Nagumo equations. In this paper, we prove that the fast pulses with oscillatory tails are also nonlinearly stable. Similar to the case of monotone tails, stability is decided by the location of a nontrivial eigenvalue near the origin of the PDE linearization about the traveling pulse. We prove that this real eigenvalue is always negative. However, the expression that governs the sign of this eigenvalue for oscillatory pulses differs from that for monotone pulses, and we show indeed that the nontrivial eigenvalue in the monotone case scales with $$\varepsilon $$ , while the relevant scaling in the oscillatory case is $$\varepsilon ^{2/3}$$ .

Stochastic Approximations and Perturbations in Forward-Backward Splitting for Monotone Operators ∗

Abstract: We investigate the asymptotic behavior of a stochastic version of the forward-backward splitting algorithm for finding a zero of the sum of a maximally monotone set-valued operator and a cocoercive operator in Hilbert spaces. Our general setting features stochastic approximations of the cocoercive operator and stochastic perturbations in the evaluation of the resolvents of the set-valued operator. In addition, relaxations and not necessarily vanishing proximal parameters are allowed. Weak and strong almost sure convergence properties of the iterates is established under mild conditions on the underlying stochastic processes. Leveraging these results, we also establish the almost sure convergence of the iterates of a stochastic variant of a primal-dual proximal splitting method for composite minimization problems.

Finite Difference Hermite WENO Schemes for Conservation Laws, II: An Alternative Approach

Abstract: In Liu and Qiu (J Sci Comput 63:548---572, 2015), we presented a class of finite difference Hermite weighted essentially non-oscillatory (HWENO) schemes for conservation laws, in which the reconstruction of fluxes is based on the usual practice of reconstructing the flux functions. In this follow-up paper, we present an alternative formulation to reconstruct the numerical fluxes, in which we first use the solution and its derivatives directly to interpolate point values at interfaces of computational cells, then we put the point values at interface of cell in building block to generate numerical fluxes. The building block can be arbitrary monotone fluxes. Comparing with Liu and Qiu (2015), one major advantage is that arbitrary monotone fluxes can be used in this framework, while in Liu and Qiu (2015) the traditional practice of reconstructing flux functions can be applied only to smooth flux splitting. Furthermore, these new schemes still keep the effectively narrower stencil of HWENO schemes in the process of reconstruction. Numerical results for both one and two dimensional equations including Euler equations of compressible gas dynamics are provided to demonstrate the good performance of the methods.

Convergent semi-Lagrangian methods for the Monge-Amp\`ere equation on unstructured grids

Abstract: This paper is concerned with developing and analyzing convergent semi-Lagrangian methods for the fully nonlinear elliptic Monge-Amp\`ere equation on general triangular grids. This is done by establishing an equivalent (in the viscosity sense) Hamilton-Jacobi-Bellman formulation of the Monge-Amp\`ere equation. A significant benefit of the reformulation is the removal of the convexity constraint from the admissible space as convexity becomes a built-in property of the new formulation. Moreover, this new approach allows one to tap the wealthy numerical methods, such as semi-Lagrangian schemes, for Hamilton-Jacobi-Bellman equations to solve Monge-Amp\`ere type equations. It is proved that the considered numerical methods are monotone, pointwise consistent and uniformly stable. Consequently, its solutions converge uniformly to the unique convex viscosity solution of the Monge-Amp\`ere Dirichlet problem. A super-linearly convergent Howard's algorithm, which is a Newton type method, is utilized as the nonlinear solver to take advantage of the monotonicity of the scheme. Numerical experiments are also presented to gauge the performance of the proposed numerical method and the nonlinear solver.

Pitt's inequalities and uncertainty principle for generalized Fourier transform

Abstract: We study the two-parameter family of unitary operators \[ \mathcal{F}_{k,a}=\exp\Bigl(\frac{i\pi}{2a}\,(2\langle k\rangle+{d}+a-2 )\Bigr) \exp\Bigl(\frac{i\pi}{2a}\,\Delta_{k,a}\Bigr), \] which are called $(k,a)$-generalized Fourier transforms and defined by the $a$-deformed Dunkl harmonic oscillator $\Delta_{k,a}=|x|^{2-a}\Delta_{k}-|x|^{a}$, $a>0$, where $\Delta_{k}$ is the Dunkl Laplacian. Particular cases of such operators are the Fourier and Dunkl transforms. The restriction of $\mathcal{F}_{k,a}$ to radial functions is given by the $a$-deformed Hankel transform $H_{\lambda,a}$. We obtain necessary and sufficient conditions for the weighted $(L^{p},L^{q})$ Pitt inequalities to hold for the $a$-deformed Hankel transform. Moreover, we prove two-sided Boas--Sagher type estimates for the general monotone functions. We also prove sharp Pitt's inequality for $\mathcal{F}_{k,a}$ transform in $L^{2}(\mathbb{R}^{d})$ with the corresponding weights. Finally, we establish the logarithmic uncertainty principle for $\mathcal{F}_{k,a}$.

Combinatorial Prophet Inequalities

Abstract: We introduce a novel framework of Prophet Inequalities for combinatorial valuation functions. For a (non-monotone) submodular objective function over an arbitrary matroid feasibility constraint, we give an $O(1)$-competitive algorithm. For a monotone subadditive objective function over an arbitrary downward-closed feasibility constraint, we give an $O(\log n \log^2 r)$-competitive algorithm (where $r$ is the cardinality of the largest feasible subset). Inspired by the proof of our subadditive prophet inequality, we also obtain an $O(\log n \cdot \log^2 r)$-competitive algorithm for the Secretary Problem with a monotone subadditive objective function subject to an arbitrary downward-closed feasibility constraint. Even for the special case of a cardinality feasibility constraint, our algorithm circumvents an $\Omega(\sqrt{n})$ lower bound by Bateni, Hajiaghayi, and Zadimoghaddam \cite{BHZ13-submodular-secretary_original} in a restricted query model. En route to our submodular prophet inequality, we prove a technical result of independent interest: we show a variant of the Correlation Gap Lemma for non-monotone submodular functions.