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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]

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]

Given a closed convex set A in a Banach space X, this paper considers the continuity and differentiability of A. The continuity of a closed convex set was introduced and studied by Gale and Klee [Math. Scand., 7 (1959), pp. 370--391] in terms of its support functional, and the differentiability of a closed convex set is a new notion introduced again in terms of its support functional. Using the technique of variational analysis, we prove that A is differentiable if and only if for every continuous linear (or convex) function f : X → R bounded below on A the corresponding optimization problem infxεA f(x) is well-posed solvable. In the reflexive space case, we prove that A is continuous if and only if for every continuous linear (or convex) function f : X → R bounded below on A the corresponding optimization problem infxεAf(x) is weakly well-posed solvable. We also prove that if the conjugate function f* of a given continuous convex function f on X is Fréchet differentiable (resp., continuous) on dom(f*), then for every closed convex set K in X with infxεKf(x) > - ∞ the corresponding optimization problem with objective f and constraint set K is well-posed (resp., weakly well-posed) solvable. In the framework of finite-dimensional spaces, several sharper results are established. [ABSTRACT FROM AUTHOR]

We characterize the subdifferential of the supremum function of finitely and infinitely indexed families of convex functions. The main contribution of this paper consists of providing formulas for such a subdifferential under weak continuity assumptions. The resulting formulas are given in terms of the exact subdifferential of the data functions at the reference point, and not at nearby points as in [Valadier, C. R. Math. Acad. Sci. Paris, 268 (1969), pp. 39--42]. We also derive new Fritz John- and KKT-type optimality conditions for semi-infinite convex optimization, omitting the continuity/closedness assumptions in [Dinh et al., ESAIM Control Optim. Calc. Var., 13 (2007), pp. 580--597]. When the family of functions is finite, we use continuity conditions concerning only the active functions, and not all the data functions as in [Rockafellar, Proc. Lond. Math. Soc. (3), 39 (1979), pp. 331--355; Volle, Acta Math. Vietnam., 19 (1994), pp. 137--148]. [ABSTRACT FROM AUTHOR]

We present in this paper two different classes of general multiple-splitting algorithms for solving finite-dimensional convex optimization problems. Under the assumption that the function being minimized can be written as the sum of K convex functions, each of which has a Lipschitz continuous gradient, we prove that the number of iterations needed by the first class of algorithms to obtain an ε-optimal solution is O((K - 1)L/ε), where L is an upper bound on all of the Lipschitz constants. The algorithms in the second class are accelerated versions of those in the first class, where the complexity result is improved to O(√(K - 1)L/ε) while the computational effort required at each iteration is almost unchanged. To the best of our knowledge, the complexity results presented in this paper are the first ones of this type that have been given for splitting and alternating direction-type methods. Moreover, all algorithms proposed in this paper are parallelizable, which makes them particularly attractive for solving certain large-scale problems. [ABSTRACT FROM AUTHOR]

In this paper, we propose an efficient approach for solving a class of large-scale convex optimization problems. The problem we consider is the minimization of a convex function over a simple (possibly infinite-dimensional) convex set, under the additional constraint Au ∈ T, where A is a linear operator and T is a convex set whose dimension is small compared to the dimension of the feasible region. In our approach, we dualize the linear constraints, solve the resulting dual problem with a purely dual gradient-type method and show how to reconstruct an approximate primal solution. Because the linear constraints have been dualized, the dual objective function typically becomes separable, mid therefore easy to compute. In order to accelerate our scheme, we introduce a novel double smoothing technique that involves regularization of the dual problem to allow the use of a fast gradient method. As a result, we obtain a method with complexity O(1/ε ln 1/ε) gradient iterations, where ε is the desired accuracy for the primal-dual solution. Our approach covers, in particular, optimal control problems with a trajectory governed by a system of linear differential equations, where the additional constraints can, for example, force the trajectory to visit some convex sets at certain moments in time. [ABSTRACT FROM AUTHOR]

To provide generalized solutions if a given problem admits no actual solution is an important task in mathematics and the natural sciences. It has a rich history dating back to the early 19th century, when Carl Friedrich Gauss developed the method of least squares of a system of linear equations--its solutions can be viewed as fixed points of averaged projections onto hyperplanes. A powerful generalization of this problem is to find fixed points of averaged resolvents (i.e., firmly nonexpansive mappings). This paper concerns the relationship between the set of fixed points of averaged resolvents and certain fixed point sets of compositions of resolvents. It partially extends recent work for two mappings on a question of C. Byrne. The analysis suggests a reformulation in a product space. Algorithmic consequences are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider both a variant of Tseng's modified forward backward splitting method and all extension of Korpelevich's method for solving hemivariational inequalities with Lipschitz continuous operators. By showing that these methods are special cases of the hybrid proximal extra gradient method introduced by Solodov and Svaiter, we derive iteration-complexity bounds for them to obtain different types of approximate solutions. In the context of saddle-point problems, we also derive complexity bounds for these methods to obtain another type of an approximate solution, namely, that of an approximate saddle point. Finally, we illustrate the usefulness of the above results by applying them to a large class of linearly constrained convex programming problems, including, for example, cone programming and problems whose objective functions converge to infinity as the Boundaries of their effective domains are approached. [ABSTRACT FROM AUTHOR]

The question of determining strong convex underestimators for nonlinear functions is theoretically and practically of major interest. Unfortunately, results along these lines are quite limited as very few general procedures are at hand that can be applied to general classes of functions. In this paper we show how to reduce the question of determining a convex envelope to lower-dimensional optimization problems when the underlying function is indefinite and (n-1)-convex. Our structural result about this reduction technique enables us to give descriptions for the convex envelope of a variety of two-dimensional functions. [ABSTRACT FROM AUTHOR]

Abstract. In this paper we compute explicit solutions of the eigenvalue problem -div(Du/|Du|) = u in R2, in particular explicit solutions whose truncatures are in Wloc1,1 (R2),and piecewise constant ones which are sums of characteristic functions of convex sets. The solutions of the above eigenvalue problem describe the asymptotic behavior of solutions of the minimizing total variation flow. As an application,we also construct explicit solutions of the denoising problem in image processing. [ABSTRACT FROM AUTHOR]

This paper investigates the complexity of steepest descent algorithms for two classes of discrete convex functions: M-convex functions and L-convex functions. Simple tie-breaking rules yield complexity bounds that are polynomials in the dimension of the variables and the size of the effective domain. Combining the present results with a standard scaling approach leads to an efficient algorithm for L-convex function minimization. [ABSTRACT FROM AUTHOR]

A transshipment problem with demands that exceed network capacity can be solved by sending flow in several waves. How can this be done in the minimum number, T, of waves, and at minimum cost, if costs are piecewise linear convex functions of the flow? In this paper, we show that this problem can be solved using min{m, log T, log(mΓU) / 1+log(mΓU)-log(mU)} maximum flow computations and one minimum (convex) cost flow computation. Here m is the number of arcs, Γ is the maximum supply or demand, and U is the maximum capacity. When there is only one sink, this problem can be solved in the same asymptotic time as one minimum (convex) cost flow computation. This improves upon the previous best algorithm to solve the problem without costs by a factor of k. Our solutions start with a stationary fractional flow and use rounding to transform this into an integral flow. The rounding procedure takes O(n) time. [ABSTRACT FROM AUTHOR]