Search

This paper provides a probabilistic proof of the comparison result for viscosity solutions of path-dependent semilinear PDEs. We consider the notion of viscosity solutions introduced in [I. Ekren, et al., Ann. Probab., 42 (2014), pp. 204-236], which considers as test functions all those smooth processes which are tangent in mean. When restricted to the Markovian case, this definition induces a larger set of test functions and reduces to the notion of stochastic viscosity solutions analyzed in [E. Bayraktar and M. Sirbu, Proc. Amer. Math. Soc., 140 (2012), pp. 3645-3654; SIAM J. Control Optim., 51 (2013), pp. 4274-4294]. Our main result takes advantage of this enlargement of the test functions and provides an easier proof of comparison. This is most remarkable in the context of the linear path-dependent heat equation. As a key ingredient for our methodology, we introduce a notion of punctual differentiation, similar to the corresponding concept in the standard viscosity solutions [L. A. Caffarelli and X. Cabre, Amer. Math. Soc. Colloq. Publ., 43, AMS, Providence, RI, 1995], and we prove that semimartingales are almost everywhere punctually differentiable. This smoothness result can be viewed as the counterpart of the Aleksandroff smoothness result for convex functions. A similar comparison result was established earlier in [I. Ekren et al., Ann. Probab., 42 (2014), pp. 204-236]. The result of this paper is more general and, more importantly, the arguments that we develop do not rely on any representation of the solution. [ABSTRACT FROM AUTHOR]

We analyze the maximal growth of trajectories of discrete-time linear switching system, i.e., controlled linear systems with the control set being an arbitrary compact set of matrices. This is done by applying the optimal convex Lyapunov function called the Barabanov norm, which provides a very refined analysis of trajectories. Until recently that notion remained rather theoretical apart from special cases. In 2015 N. Guglielmi and M. Zennaro [SIAM J. Matrix Anal. Appl., 36 (2015), pp. 634-655] showed that many systems possess at least one efficiently computed Barabanov norm. In this paper we classify all possible Barabanov norms and prove that, under mild assumptions, which can be verified algorithmically, those norms are unique and are either piecewise-linear or piecewise-quadratic. For some narrow classes of systems, there are more complicated Barabanov norms, but they can still be classified and constructed. Using those results we find all trajectories of the fastest growth. Examples and numerical results are provided. [ABSTRACT FROM AUTHOR]

In this paper we consider a distributed stochastic optimization problem without gradient/subgradient information for local objective functions and subject to local convex constraints. Objective functions may be nonsmooth and observed with stochastic noises, and the network for the distributed design is time-varying. By adding stochastic dithers to local objective functions and constructing randomized differences motivated by the Kiefer--Wolfowitz algorithm, we propose a distributed subgradient-free algorithm for finding the global minimizer with local observations. Moreover, we prove that the consensus of estimates and global minimization can be achieved with probability one over the time-varying network, and we obtain the convergence rate of the mean average of estimates as well. Finally, we give numerical examples to illustrate the performance of the proposed algorithms. [ABSTRACT FROM AUTHOR]

The joint spectral radius (JSR) of a set of matrices characterizes the maximal asymptotic growth rate of an infinite product of matrices of the set. This quantity appears in a number of applications including the stability of switched and hybrid systems. A popular method used for the stability analysis of these systems searches for a Lyapunov function with convex optimization tools. We investigate dual formulations for this approach and leverage these dual programs for developing new analysis tools for the JSR. We show that the dual of this convex problem searches for the occupations measures of trajectories with high asymptotic growth rate. We both show how to generate a sequence of guaranteed high asymptotic growth rate and how to detect cases where we can provide lower bounds on the JSR. All results of this paper are presented for the general case of constrained switched systems, that is, systems for which the switching signal is constrained by an automaton. [ABSTRACT FROM AUTHOR]

We study the following optimization problem over a dynamical system that consists of several linear subsystems: Given a finite set of nÃ--n matrices and an n-dimensional vector, find a sequence of K matrices, each chosen from the given set of matrices, to maximize a convex function over the product of the K matrices and the given vector. This simple problem has many applications in operations research and control, yet a moderate-sized instance is challenging to solve to optimality for state-of-the-art optimization software. We propose a simple exact algorithm for this problem. Our algorithm runs in polynomial time when the given set of matrices has the oligo-vertex property, a concept we introduce in this paper for a finite set of matrices. We derive several sufficient conditions for a set of matrices to have the oligo-vertex property. Numerical results demonstrate the clear advantage of our algorithm in solving large-sized instances of the problem over one state-of-the-art global optimization solver. We also propose several open questions on the oligo-vertex property and discuss its potential connection with the finiteness property of a set of matrices, which may be of independent interest. [ABSTRACT FROM AUTHOR]

This paper studies distributed algorithms for the nonsmooth extended monotropic optimization problem, which is a general convex optimization problem with a certain separable structure. The considered nonsmooth objective function is the sum of local objective functions as-signed to agents in a multiagent network, with local set constraints and affine equality constraints. Each agent only knows its local objective function, local set constraint, and the information ex-changed between neighbors. To solve the constrained convex optimization problem, we propose two novel distributed continuous-time subgradient-based algorithms, with projected output feedback and derivative feedback, respectively. Moreover, we prove the convergence of proposed algorithms to the optimal solutions under some mild conditions and analyze convergence rates, with the help of the techniques of variational inequalities, decomposition methods, and differential inclusions. Finally, we give an example to illustrate the efficacy of the proposed algorithms. [ABSTRACT FROM AUTHOR]

The paper studies convex Lyapunov functions for differential and difference inclusions with right-hand sides given by convex processes, that is, by set-valued mappings the graphs of which are convex cones. Convex conjugacy between weak Lyapunov functions for such inclusions and Lyapunov functions for adjoint inclusions is established. Asymptotic stability concepts are compared, and the existence of convex Lyapunov functions for classes of convex processes is shown. The relevance of the results for the study of asymptotic controllability or stabilizability and of detectability for linear control systems with conical control and state constraints is underlined. [ABSTRACT FROM AUTHOR]

This paper discusses asymptotic analysis and adaptive design of convex estimators over the Hölder class under the sup-norm risk and the pointwise risk using constrained optimization and asymptotic statistical techniques. Specifically, convex B-spline estimators are proposed to achieve uniform optimal convergence rates and adaptive procedures. The presence of the convex shape constraint complicates asymptotic performance analysis, particularly uniform convergence analysis. This in turn requires deep understanding of a family of size varying constrained optimization problems on spline coeficients. To address these issues, we establish the uniform Lipschitz property of optimal spline coeficients in the l∞-norm by exploiting the structure of underlying constrained optimization 8 problems. By using this property, polyhedral theory, and statistical techniques, we show that the convex B-spline estimator attains uniform consistency and optimal rates of convergence on the entire interval of interest over the Hölder class under the sup-norm risk and the pointwise risk. In addition, adaptive estimates are constructed under both risks when the Hölder exponent is between one and two. These estimates achieve a maximal risk within a constant factor of the minimax risk over the Hölder class. [ABSTRACT FROM AUTHOR]

In this paper, we study the approximation of a Dirichlet control problem governed by an elliptic equation defined on a curved domain ω. To solve this problem numerically, it is usually necessary to approximate Ω by a (typically polygonal) new domain Ωh. The difference between the solutions of both infinite-dimensional control problems, one formulated in Ω and the second in Ωh, was studied ill [E. Casas and J. Sokolowski, SIAM J. Control Optim., 48 (2010), pp. 3746-3780], where an error of order O(h) was proved. In [K. Deckelnick, A. Günther, and M. Hinze, SIAM J. Control Optim., 48 (2009), pp. 2798-2819], the numerical approximation of the problem defined in Ω was considered. The authors used a finite element method such that Ωh was the polygon formed by the union of all triangles of the mesh of parameter h. They proved an error of order O(h3/2) for the difference between continuous and discrete optimal controls. Here we show that the estimate obtained in [E. Casas and J. Sokolowski, SIAM J. Control Optim., 48 (2010), pp. 3746-3780] cannot be improved, which leads to the paradox that the numerical solution is a better approximation of the optimal control than the exact one obtained just by changing the domain from Ω to Ωh. [ABSTRACT FROM AUTHOR]

This paper is focused on a cooperative control design which guarantees robust connectivity and invariance of a multiagent network inside a bounded domain, under the presence of additional bounded input terms in each agent's dynamics. In particular, under the assumptions that the domain is convex and has a smooth boundary, we can design a repulsion vector field near its boundary, which ensures invariance of the agents' trajectories and does not affect the robustness properties of the control part that is exploited for connectivity maintenance. [ABSTRACT FROM AUTHOR]

In this paper we provide a solution to the gain-scheduling synthesis problem (GSP) for structured parametric and dynamic passive uncertainties, which encompasses an approach for norm-bounded uncertainties as considered in classical structured singular value theory. If compared to known approaches, the distinguishing feature is the use of frequency-dependent multipliers to guarantee stability and performance of the controlled system. On the basis of suitable parameterizations of these multipliers, we develop necessary and sufficient conditions for the existence of gain-scheduling controllers and discuss the benefits of this new design methodology. [ABSTRACT FROM AUTHOR]

This paper deals with the analysis of a class of nonsmooth robust controllers for Lagrangian systems with nontrivial mass matrix. First the existence and uniqueness of solutions are analyzed, then the Lyapunov stability, the Krasovskii-LaSalle invariance principle, and finite-time convergence properties are studied. [ABSTRACT FROM AUTHOR]

This paper considers a numerical method for reconstructing piecewise constant Robin coefficients from boundary measurements. A novel variational formulation of the inverse problem based on the celebrated Modica-Mortola functional is proposed, and its Γ-convergence to the formulation with total variation regularization is shown. The differentiability of the functional is investigated, based on which the concave convex procedure is justified for its numerical solution. The finite element method is utilized for discretizing the functional, and the convergence of the finite element approximation is established. Three sets of numerical experiments are presented to illustrate its distinct features. [ABSTRACT FROM AUTHOR]

We study some continuous dynamical systems associated with constrained optimization problems. For that purpose, we introduce the concept of elliptic barrier operators and develop a unified framework to derive and analyze the associated class of gradient-like dynamical systems, called A-driven descent method (A-DM). Prominent methods belonging to this class include several continuous descent methods studied earlier in the literature such as steepest descent method, continuous gradient projection methods and Newton-type methods as well as continuous interior descent methods such as Lotka-Volterra-type differential equations and Riemannian gradient methods. Related discrete iterative methods such as proximal interior point algorithms based on Bregman functions and second order homogeneous kernels can also be recovered within our framework and allow for deriving some new and interesting dynamics. We prove global existence and strong viability results of the corresponding trajectories of (A-DM) for a smooth objective function. When the objective function is convex, we analyze the asymptotic behavior at infinity of the trajectory produced by the proposed class of dynamical systems (A -DM). In particular, we derive a general criterion ensuring the global convergence of the trajectory of (A-DM) to a minimizer of a convex function over a closed convex set. This result is then applied to several dynamics built upon specific elliptic barrier operators. Throughout the paper, our results are illustrated with many examples. [ABSTRACT FROM AUTHOR]

The relationship of the theory of port-Hamiltonian systems with the mathematical concept of monotonicity is explored. The earlier introduced notion of incrementally port-Hamiltonian systems is extended to systems defined with respect to maximal cyclically monotone relations, together with their generating convex functions. This gives rise to interesting subclasses of incrementally port-Hamiltonian systems, with examples stemming from physical systems modeling as well as from convex optimization. Furthermore, it is shown how cyclical monotonicity for Dirac structures is equivalent to separability. An in-depth treatment is given of the composition of maximal monotone and maximal cyclically monotone relations, where in the latter case the resulting maximal cyclically monotone relation is shown to be computable through the use of generating functions. The results on compositionality are employed for steady-state analysis and for a convex optimization approach to the computation of the equilibria of interconnected incrementally port-Hamiltonian systems. Finally, the relation to incremental and differential passivity is discussed, and it is shown how incrementally port-Hamiltonian systems with strictly convex Hamiltonians are equilibrium independent passive. [ABSTRACT FROM AUTHOR]

We present a convex solution for the design of generalized accelerated gradient algorithms for strongly convex objective functions with Lipschitz continuous gradients. We utilize integral quadratic constraints and the Youla parameterization from robust control theory to formulate a solution of the algorithm design problem as a convex semidefinite program. We establish explicit formulas for the optimal convergence rates and extend the proposed synthesis solution to extremum control problems. [ABSTRACT FROM AUTHOR]

For a locally Lipschitz integral functional If on L¹(T, Rn associated with a measurable integrand f, the limiting subdifferential and the approximate subdifferential never coincide at a point x0where f(t, ⋅) is not subdifferentially regular at x0 (t) for a.e. t Ε T. The coincidence of both subdifferentials occurs on a dense set of L¹(T, Rn) if and only if f(t, ⋅) is convex for a.e. t Ε T. Our results allow us to characterize Aubin's Lipschitz-like property as well as the convexity of multivalued mappings between L¹-spaces. New necessary optimality conditions for some Bolza problems are also obtained. [ABSTRACT FROM AUTHOR]

In a real Hilbert space setting, we investigate the asymptotic properties of the solutions of a second-order differential system in view of linearly constrained convex minimization. The inertial dynamics are governed by a Hessian-driven damping term associated with the convex function to be minimized and potential effects induced by the linear constraints. We provide conditions on both the damping and the potential for which the solutions converge towards some feasible point of the convex minimization problem; this convergence is towards some minimizer provided that the solutions' initial data is specifically preselected. In addition, we present asymptotic estimates on the convergence rate of the solutions depending on the interaction between damping and potential effects. Our analysis is mainly based on energy-like arguments that capture the dissipative nature of the inertial dynamics by means of a Bregman distance. We complement our study with the fact that the second-order dynamics admit a first-order representation in terms of the Arrow--Hurwicz differential system. Numerical experiments further illustrate our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper we develop and present new convex synthesis conditions for robust performance in linear time-varying systems, subject to time-varying perturbations. In particular, these results are exact for sensitivity minimization in the presence of multiplicative perturbations. The methods apply to a number of additional robust performance problems and are always both necessary and sufficient when the system plant is periodic. Otherwise, the conditions provided are always sufficient, and a controller can be directly constructed when this condition holds. [ABSTRACT FROM AUTHOR]

Our aim is to present sufficient conditions ensuring Hoffman's error bound for lower semicontinuous nonconvex inequality systems and to analyze its impact on the local controllability, implicit function theorem for (non-Lipschitz) multivalued mappings, generalized equations (variational inequalities), and sensitivity analysis and on other problems like Lipschitzian properties of polyhedral multivalued mappings as well as weak sharp minima or linear conditioning. We show how the information about our sufficient conditions can be used to provide a computable constant such that Hoffman's error bound holds. We also show that this error bound is nothing but the classical Farkas lemma for linear inequality systems. In the latter case our constant may be computed explicitly. [ABSTRACT FROM AUTHOR]

We prove that the Lavrentiev phenomenon does not occur for the functional ∫Ω[l(|∇ u(x)|) + G(u)] dx on u - u0 ∈ W01, 1 (Ω), where L and G are convex functions subject to additional regularity assumptions, and where ∂Ω and u0 are smooth. [ABSTRACT FROM AUTHOR]

The formulation and the semismooth Newton solution of Nash equilibrium multiobjective elliptic optimal control problems are presented. Existence and uniqueness of a Nash equilibrium are proved. The corresponding solution is characterized by an optimality system that is approximated by second-order finite differences and solved with a semismooth Newton scheme. It is demonstrated that the numerical solution is second-order accurate and that the semismooth Newton iteration is globally and locally quadratically convergent. Results of numerical experiments confirm the theoretical estimates and show the effectiveness of the proposed computational framework. [ABSTRACT FROM AUTHOR]

We present a data structure for answering approximate shortest path queries in a planar subdivision from a fixed source. Let ρ ⩾ 1 be a real number. Distances in each face of this subdivision are measured by a possibly asymmetric convex distance function whose unit disk is contained in a concentric unit Euclidean disk and contains a concentric Euclidean disk with radius 1/ρ. Different convex distance functions may be used for different faces, and obstacles are allowed. Let ϵ be any number strictly between 0 and 1. Our data structure returns a (1+ϵ) approximation of the shortest path cost from the fixed source to a query destination in O(log ρn/ϵ ) time. Afterwards, a (1 +ϵ)-approximate shortest path can be reported in O(log n) time plus the complexity of the path. The data structure uses O( ρ²n³/ ϵ² log ρn/ϵ ) space and can be built in O( ρ²n³/ϵ² (log ρn/ϵ)²) time. Our time and space bounds do not depend on any other parameter; in particular, they do not depend on any geometric parameter of the subdivision such as the minimum angle. [ABSTRACT FROM AUTHOR]

The celebrated Brezis-Ekeland principle [C. R. Acad. Sci. Paris Ser. A-B, 282 (1976), pp. Ai, A1197-A1198, Aii, and A971-A974] characterizes trajectories of nonautonomous gradient flows of convex functionals as solutions to suitable minimization problems. This note extends this characterization to doubly nonlinear evolution equations driven by convex potentials. The characterization is exploited in order to establish approximation results for gradient flows, doubly nonlinear equations, and rate-independent evolutions. [ABSTRACT FROM AUTHOR]

A fundamental result due to Rockafellar states that every cyclically monotone operator A admits an antiderivative f in the sense that the graph of A is contained in the graph of the subdifferential operator ∂f. Given a method m that assigns every finite cyclically monotone operator A some antiderivative mA, we say that the method is primal-dual symmetric if m applied to the inverse of A produces the Fenchel conjugate of mA. Rockafellar's antiderivatives do not possess this property. Utilizing Fitzpatrick functions and the proximal average, we present novel primal-dual symmetric intrinsic methods. The antiderivatives produced by these methods provide a solution to a problem posed by Rockafellar in 2005. The results leading to this solution are illustrated by various examples. [ABSTRACT FROM AUTHOR]

The gap between the complex structured singular value of a complex matrix M and its convex upper bound is considered. New necessary and sufficient conditions for the existence of the gap are derived. It is shown that determining whether there exists such a gap is as difficult as evaluating a structured singular value of a reduced rank matrix (whose rank is equal to the multiplicity of the largest singular value of M). Furthermore, if an upper bound on this reduced rank problem can be obtained, it is shown that this provides an upper bound on the original problem that is lower than the convex relaxation upper bound. An example that illustrates our procedure is given. We also give the solution of several structured-approximation problems of independent interest. [ABSTRACT FROM AUTHOR]

The classical study of controllability of linear systems assumes unconstrained control inputs. The ‘distance to uncontrollability’ measures the size of the smallest perturbation to the matrix description of the system rendering it uncontrollable and is a key measure of system robustness. We extend the standard theory of this measure of controllability to the case where the control input must satisfy given linear inequalities. Specifically, we consider the control of differential inclusions, concentrating on the particular case where the control input takes values in a given convex cone. [ABSTRACT FROM AUTHOR]

We study the existence of Lipschitz minimizers of integral functionals $$ \mathcal{I}(u)=\int_{\Omega} \varphi(x,\textrm{det}\,Du(x))\,dx,$$ where $\Omega$ is an open subset of $\mathbb{R}^N$ with Lipschitz boundary, $\varphi:\Omega\times (0,+\infty)\to [0,+\infty)$ is a continuous function, and $u\in W^{1,N}(\Omega, \mathbb{R}^N)$, $u(x)=x$ on $\partial \Omega$. We consider both the cases of $\varphi$ convex and nonconvex with respect to the last variable. The attainment results are obtained passing through the minimization of an auxiliary functional and the solution of a prescribed Jacobian equation. [ABSTRACT FROM AUTHOR]

A nonconvex problem of constrained optimization is analyzed in terms of its ordinary Lagrangian function. New sufficient conditions are obtained for the duality gap to vanish. Among them, the main condition is that the objective and constraint functions be the sums of convex functionals and nonconvex quadratic forms with certain specific spectral properties. The proofs are related to extensions of the classic Toeplitz-Hausdorff theorem, which states that a continuous quadratic mapping (y1, y2) = [B[SUB1](z), B[SUB2](z)] from a complex Hilbert space H = {z} into R[SUP2] = {(y1,y2)} transforms the unit sphere |z| = 1 into a convex set. The extensions deal with a quadratic mapping [B[SUB1](z), , B[SUBk](z)] from a real Hilbert space into R[SUPk] with k being arbitrary. Applications to linear-quadratic optimal control theory are considered. [ABSTRACT FROM AUTHOR]