Another efficient algorithm for convex hulls in two dimensions

Andrew, Alex M.

doi:10.1016/0020-0190(79)90072-3

Cited by 383 publications

(204 citation statements)

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“…If d = 1 and F is finite and given in increasing order along the first coordinate, then c(ψ) can be calculated in time proportional to the size of F . This is because the upper hull of a sorted set of planar points (Andrew 1979) can be calculated in linear time, as described in proposition 3.3. Let x and x be distinct elements of Im(X 1 ) such that (X 0 , c(ψ)) belongs to the segment joining (x , ψ * (x )) to (x , ψ * (x )).…”

Section: Practical Implementation Issuesmentioning

confidence: 99%

Superreplication of Financial Derivatives via Convex Programming

Kahale

2017

Management Science

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We give a method based on convex programming to calculate the optimal super-replicating and sub-replicating prices and corresponding hedging strategies of a financial derivative in terms of other financial derivatives. Our method finds a model that matches the superreplicating (or sub-replicating) price within an arbitrary precision and is consistent with the other financial derivatives prices. Applications include robust replication in terms of call prices with various strikes and maturities of forward start options, volatility and variance swaps and derivatives, cliquets calls, barrier options, lookback and Asian options. Numerical examples show that, in some cases, the super-replicating and/or sub-replicating prices are within 10% of the Black-and-Scholes price but considerably differ from it in other cases. Our method can take into account bid-ask spreads, interest rates and dividends and various limitations to the diffusion model. An alternative method to optimally super-replicate and sub-replicate forward start options using semi-definite and linear programming is presented.

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Section: Practical Implementation Issuesmentioning

confidence: 99%

Superreplication of Financial Derivatives via Convex Programming

Kahale

2017

Management Science

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“…Since then numerous improvements [1,2,3,15,41] and extensions to polytopes [53,74] have been proposed. An overview can be found in [54,60].…”

Section: Convex Hullmentioning

confidence: 99%

The two variable per inequality abstract domain

Simon

King

2010

Higher-Order Symb Comput

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Abstract. This article presents the Two Variable Per Inequality abstract domain (TVPI domain for short). This so-called weakly-relational domain is able to express systems of linear inequalities where each inequality has at most two variables. The domain represents a sweet-point in the performance-cost tradeoff between the faster Octagon domain and the more expressive domain of general convex polyhedra. In particular, we detail techniques to closely approximate integral TVPI systems, thereby finessing the problem of excessively growing coefficients, yielding -to our knowledge -the only relational domain that combines linear relations with arbitrary coefficients and strongly polynomial performance. Keywords: polyhedral analysis, integer programming, abstract interpretationStatic analysis methods have evolved from inferring prerequisite invariants for compiler optimisations to tools in their own right that are able to prove the absence of run-time errors of software. The abstract interpretation framework [21] provides a way of constructing and justifying program analyses. The key idea is to simulate each operation in a program with an abstract analog that operates on a description of the program state, rather than the state itself. The descriptions are chosen to trace a property of interest, for instance, an interval that encloses all possible values of a variable would be of value when reasoning about array bounds. These descriptions constitute what is known as an abstract domain. An abstract domain is usually presented as a lattice D, , , and the analysis itself is formulated as a set of recursive equations over D. The recursive equations are solved iteratively, until a fixpoint is reached which is detected using the ordering predicate . Each equation expresses how a program statement transforms the state at one program point to the state at another. Equations may be recursive because of loops in the program. The meet and join † This paper is a revised extract of the first author's PhD thesis [65], which, in turn, extends [72].

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“…In this section, we present a simple in-place implementation of Graham's convex hull algorithm [1] or, more precisely, Andrew's modification of Graham's algorithm [3]. The algorithm requires the use of an in-place sorting algorithm.…”

Section: An O(n Log N) Time Algorithmmentioning

confidence: 99%

“…Theorem 2 Opt-Graham-InPlace-Hull computes the convex hull of n points in O(n log n) time using at most 3n−h right turn tests, 3 2 n log 2 n+O(n) swaps, n log 2 n+O(n) lexicographic comparisons and O(1) additional memory, where h is the number of vertices of the convex hull.…”

Section: The Optimized Algorithmmentioning

confidence: 99%