1,465 research outputs found
A Sequential Quadratic Programming Method for Volatility Estimation in Option Pricing
Our goal is to identify the volatility function in Dupire's equation from given option prices. Following an optimal control approach in a Lagrangian framework, we propose a globalized sequential quadratic programming (SQP) algorithm with a modified Hessian - to ensure that every SQP step is a descent direction - and implement a line search strategy. In each level of the SQP method a linear-quadratic optimal control problem with box constraints is solved by a primal-dual active set strategy. This guarantees L1 constraints for the volatility, in particular assuring its positivity. The proposed algorithm is founded on a thorough first- and second-order optimality analysis. We prove the existence of local optimal solutions and of a Lagrange multiplier associated with the inequality constraints. Furthermore, we prove a sufficient second-order optimality condition and present some numerical results underlining the good properties of the numerical scheme.Dupire equation, parameter identification, optimal control, optimality conditions, SQP method, primal-dual active set strategy
Invariant Synthesis for Incomplete Verification Engines
We propose a framework for synthesizing inductive invariants for incomplete
verification engines, which soundly reduce logical problems in undecidable
theories to decidable theories. Our framework is based on the counter-example
guided inductive synthesis principle (CEGIS) and allows verification engines to
communicate non-provability information to guide invariant synthesis. We show
precisely how the verification engine can compute such non-provability
information and how to build effective learning algorithms when invariants are
expressed as Boolean combinations of a fixed set of predicates. Moreover, we
evaluate our framework in two verification settings, one in which verification
engines need to handle quantified formulas and one in which verification
engines have to reason about heap properties expressed in an expressive but
undecidable separation logic. Our experiments show that our invariant synthesis
framework based on non-provability information can both effectively synthesize
inductive invariants and adequately strengthen contracts across a large suite
of programs
Haefliger structures and symplectic/contact structures
For some geometries including symplectic and contact structures on an
n-dimensional manifold, we introduce a two-step approach to Gromov's
h-principle. From formal geometric data, the first step builds a transversely
geometric Haefliger structure of codimension n. This step works on all
manifolds, even closed. The second step, which works only on open manifolds and
for all geometries, regularizes the intermediate Haefliger structure and
produces a genuine geometric structure. Both steps admit relative parametric
versions. The proofs borrow ideas from W. Thurston, like jiggling and
inflation. Actually, we are using a more primitive jiggling due to R. Thom.Comment: To appear in Journal de l'Ecole Polytechniqu
Simple Problems: The Simplicial Gluing Structure of Pareto Sets and Pareto Fronts
Quite a few studies on real-world applications of multi-objective
optimization reported that their Pareto sets and Pareto fronts form a
topological simplex. Such a class of problems was recently named the simple
problems, and their Pareto set and Pareto front were observed to have a gluing
structure similar to the faces of a simplex. This paper gives a theoretical
justification for that observation by proving the gluing structure of the
Pareto sets/fronts of subproblems of a simple problem. The simplicity of
standard benchmark problems is studied.Comment: 10 pages, accepted at GECCO'17 as a poster paper (2 pages
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