SSDB spaces and maximal monotonicity

In this paper, we develop some of the theory of SSD spaces and SSDB spaces, and deduce some results on maximally monotone multifunctions on a reflexive Banach space.Comment: 16 pages. Written version of the talk given at IX ISORA in Lima, Peru, October 200

Quantum Circuits for the Unitary Permutation Problem

We consider the Unitary Permutation problem which consists, given $n$ unitary gates $U_1, \ldots, U_n$ and a permutation $\sigma$ of $\{1,\ldots, n\}$, in applying the unitary gates in the order specified by $\sigma$, i.e. in performing $U_{\sigma(n)}\ldots U_{\sigma(1)}$. This problem has been introduced and investigated by Colnaghi et al. where two models of computations are considered. This first is the (standard) model of query complexity: the complexity measure is the number of calls to any of the unitary gates $U_i$ in a quantum circuit which solves the problem. The second model provides quantum switches and treats unitary transformations as inputs of second order. In that case the complexity measure is the number of quantum switches. In their paper, Colnaghi et al. have shown that the problem can be solved within $n^2$ calls in the query model and $\frac{n(n-1)}2$ quantum switches in the new model. We refine these results by proving that $n\log_2(n) +\Theta(n)$ quantum switches are necessary and sufficient to solve this problem, whereas $n^2-2n+4$ calls are sufficient to solve this problem in the standard quantum circuit model. We prove, with an additional assumption on the family of gates used in the circuits, that $n^2-o(n^{7/4+\epsilon})$ queries are required, for any $\epsilon >0$. The upper and lower bounds for the standard quantum circuit model are established by pointing out connections with the permutation as substring problem introduced by Karp.Comment: 8 pages, 5 figure

Closedness type regularity conditions for surjectivity results involving the sum of two maximal monotone operators

In this note we provide regularity conditions of closedness type which guarantee some surjectivity results concerning the sum of two maximal monotone operators by using representative functions. The first regularity condition we give guarantees the surjectivity of the monotone operator $S(\cdot + p)+T(\cdot)$, where $p\in X$ and $S$ and $T$ are maximal monotone operators on the reflexive Banach space $X$. Then, this is used to obtain sufficient conditions for the surjectivity of $S+T$ and for the situation when $0$ belongs to the range of $S+T$. Several special cases are discussed, some of them delivering interesting byproducts.Comment: 11 pages, no figure

Set-optimization meets variational inequalities

We study necessary and sufficient conditions to attain solutions of set-optimization problems in therms of variational inequalities of Stampacchia and Minty type. The notion of a solution we deal with has been introduced Heyde and Loehne, for convex set-valued objective functions. To define the set-valued variational inequality, we introduce a set-valued directional derivative and we relate it to the Dini derivatives of a family of linearly scalarized problems. The optimality conditions are given by Stampacchia and Minty type Variational inequalities, defined both by the set valued directional derivative and by the Dini derivatives of the scalarizations. The main results allow to obtain known variational characterizations for vector valued optimization problems

Set optimization - a rather short introduction

Recent developments in set optimization are surveyed and extended including various set relations as well as fundamental constructions of a convex analysis for set- and vector-valued functions, and duality for set optimization problems. Extensive sections with bibliographical comments summarize the state of the art. Applications to vector optimization and financial risk measures are discussed along with algorithmic approaches to set optimization problems

Optimal Fair Computation

A computation scheme among n parties is fair if no party obtains the computation result unless all other n-1 parties obtain the same result. A fair computation scheme is optimistic if n honest parties can obtain the computation result without resorting to a trusted third party. We prove, for the first time, a tight lower bound on the message complexity of optimistic fair computation for n parties among which n-1 can be malicious in an asynchronous network. We do so by relating the optimal message complexity of optimistic fair computation to the length of the shortest permutation sequence in combinatorics

Unifying local-global type properties in vector optimization.

It is well-known that all local minimum points of a semistrictly quasiconvex real-valued function are global minimum points. Also, any local maximum point of an explicitly quasiconvex real-valued function is a global minimum point, provided that it belongs to the intrinsic core of the function’s domain. The aim of this paper is to show that these “local min - global min” and “local max - global min” type properties can be extended and unified by a single general localglobal extremality principle for certain generalized convex vector-valued functions with respect to two proper subsets of the outcome space. For particular choices of these two sets, we recover and refine several local-global properties known in the literature, concerning unified vector optimization (where optimality is defined with respect to an arbitrary set, not necessarily a convex cone) and, in particular, classical vector/multicriteria optimization.Nicolae Popovici’s research was supported by a grant of the Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, project number PN-III-P4-ID-PCE- 2016-0190, within PNCDI III