In this paper, we present a measure of distance in a second-order cone based on a class of continuously differentiable strictly convex functions on ₊₊. Since the distance function has some favorable properties similar to those of the D-function (Censor and Zenios in J. Optim. Theory Appl. 73:451464 [1992]), we refer to it as a quasi D-function. Then, a proximal-like algorithm using the quasi D-function is proposed and applied to the second-cone programming problem, which is to minimize a closed proper convex function with general second-order cone constraints. Like the proximal point algorithm using the D-function (Censor and Zenios in J. Optim. Theory Appl. 73:451464 [1992]; Chen and Teboulle in SIAM J. Optim. 3:538543 [1993]), under some mild assumptions we establish the global convergence of the algorithm expressed in terms of function values; we show that the sequence generated by the

We introduce an entropy-like proximal algorithm for the problem of minimizing a closed proper convex function subject to symmetric cone constraints. The algorithm is based on a distance-like function that is an extension of the Kullback-Leiber relative entropy to the setting of symmetric cones. Like the proximal algorithms for convex programming with nonnegative orthant cone constraints, we show that, under some mild assumptions, the sequence generated by the proposed algorithm is bounded and every accumulation point is a solution of the considered problem. In addition, we also present a dual application of the proposed algorithm to the symmetric cone linear program, leading to a multiplier method which is shown to possess similar properties as the exponential multiplier method (Tseng and Bertsekas in Math. Program. 60:119, 1993) holds.

Let R be a not necessarily commutative ring with 1. In the present paper we first introduce a notion of quasi-orderings, which axiomatically subsumes all the orderings and valuations on R. We proceed by uniformly defining a coarsening relation ≤ on the set Q(R) of all quasi-orderings on R. One of our main results states that (Q(R),≤′) is a rooted tree for some slight modification ≤′ of ≤, i.e. a partially ordered set admitting a maximum such that for any element there is a unique chain to that maximum. As an application of this theorem we obtain that (Q(R),≤′) is a spectral set, i.e. order-isomorphic to the spectrum of some commutative ring with 1. We conclude this paper by studying Q(R) as a topological space.

This is the second of a sequence of papers proving the quantum invariance for ordinary flops over an arbitrary smooth base. In this paper, we complete the proof of the invariance of the big quantum rings under ordinary flops of splitting type. To achieve that, several new ingredients are introduced. One is a quantum Leray--Hirsch theorem for the local model (a certain toric bundle) which extends the quantum D module of Dubrovin connection on the base by a Picard--Fuchs system of the toric fibers. Nonsplit flops as well as further applications of the quantum Leray--Hirsch theorem will be discussed in subsequent papers. In particular, a quantum splitting principle is developed in Part III which reduces the general ordinary flops to the split case solved here.

We construct an index for BPS operators supported on a ray in five dimensional superconformal field theories with exceptional global symmetries. We compute the E_n representations (for n = 2, . . . , 7) of operators of low spin, thus verifying that while the expression for the index is only SO(2n − 2) × U(1) invariant, the index itself exhibits the full E_n symmetry (at least up to the order we expanded). The ray operators we studied in 5d can be viewed as generalizations of operators constructed in a Yang-Mills theory with fundamental matter by attaching an open Wilson line to a quark. For n ≤ 7, in contrast to local operators, they carry nontrivial charge under the ℤ_{9 − n}  ⊂ E_n center of the global symmetry. The representations that appear in the ray operator index are therefore different, for n ≤ 7, from those appearing in the previously computed superconformal index. For 3 ≤ n ≤ 7, we find that the leading term in the index is a character of a minuscule representation of E_n . We also discuss the case n = 8, which presents a unique technical challenge, and remains an open problem.