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   Recent Results on Douglas–Rachford Methods for Combinatorial Optimization ProblemsAragón Artacho, Francisco Javier  ; ; in Journal of Optimization Theory and Applications (in press) We discuss recent positive experiences applying convex feasibility algorithms of Douglas-Rachford type to highly combinatorial and far from convex problems. Detailed reference viewed: 155 (16 UL)Applications of convex analysis within mathematicsAragón Artacho, Francisco Javier ; ; et alin Mathematical Programming (in press) In this paper, we study convex analysis and its theoretical applications. We first apply important tools of convex analysis to Optimization and to Analysis. We then show various deep applications of ... [more ▼] In this paper, we study convex analysis and its theoretical applications. We first apply important tools of convex analysis to Optimization and to Analysis. We then show various deep applications of convex analysis and especially infimal convolution in Monotone Operator Theory. Among other things, we recapture the Minty surjectivity theorem in Hilbert space, and present a new proof of the sum theorem in reflexive spaces. More technically, we also discuss autoconjugate representers for maximally monotone operators. Finally, we consider various other applications in mathematical analysis. [less ▲] Detailed reference viewed: 232 (24 UL)Douglas-Rachford Feasibility Methods for Matrix Completion ProblemsAragón Artacho, Francisco Javier ; ; in ANZIAM Journal (2014), 55(4), 299-326 In this paper we give general recommendations for successful application of the Douglas-Rachford reflection method to convex and non-convex real matrix-completion problems. These guidelines are ... [more ▼] In this paper we give general recommendations for successful application of the Douglas-Rachford reflection method to convex and non-convex real matrix-completion problems. These guidelines are demonstrated by various illustrative examples. [less ▲] Detailed reference viewed: 116 (1 UL)Global convergence of a non-convex Douglas-Rachford iterationAragón Artacho, Francisco Javier ; in Journal of Global Optimization (2013), 57(3), 753-769 We establish a region of convergence for the proto-typical non-convex Douglas–Rachford iteration which finds a point on the intersection of a line and a circle. Previous work on the non-convex iteration ... [more ▼] We establish a region of convergence for the proto-typical non-convex Douglas–Rachford iteration which finds a point on the intersection of a line and a circle. Previous work on the non-convex iteration Borwein and Sims (Fixed-point algorithms for inverse problems in science and engineering, pp. 93–109, 2011) was only able to establish local convergence, and was ineffective in that no explicit region of convergence could be given. [less ▲] Detailed reference viewed: 101 (10 UL)Recent Results on Douglas–Rachford MethodsAragón Artacho, Francisco Javier ; ; in Serdica Mathematical Journal (2013), 39 Recent positive experiences applying convex feasibility algorithms of Douglas–Rachford type to highly combinatorial and far from convex problems are described. Detailed reference viewed: 46 (6 UL)Walking on Real NumbersAragón Artacho, Francisco Javier ; ; et alin Mathematical Intelligencer (2013), 35(1), 42-60 Motivated by the desire to visualize large mathematical data sets, especially in number theory, we offer various tools for representing floating point numbers as planar(or three dimensional) walks and for ... [more ▼] Motivated by the desire to visualize large mathematical data sets, especially in number theory, we offer various tools for representing floating point numbers as planar(or three dimensional) walks and for quantitatively measuring their “randomness.” [less ▲] Detailed reference viewed: 51 (5 UL) |
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