By Fuad Aleskerov, Visit Amazon's Boris Goldengorin Page, search results, Learn about Author Central, Boris Goldengorin, , Panos M. Pardalos
The quantity is devoted to Boris Mirkin at the get together of his seventieth birthday. as well as his startling PhD leads to summary automata idea, Mirkin’s floor breaking contributions in numerous fields of selection making and knowledge research have marked the fourth area of the 20 th century and past. Mirkin has performed pioneering paintings in team selection, clustering, facts mining and data discovery geared toward discovering and describing non-trivial or hidden structures—first of all, clusters, orderings and hierarchies—in multivariate and/or community data.
This quantity features a number of papers reflecting fresh advancements rooted in Mirkin’s basic contribution to the state of the art in workforce selection, ordering, clustering, info mining and data discovery. Researchers, scholars and software program engineers will take advantage of new wisdom discovery suggestions and alertness directions.
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Additional info for Clusters, Orders, and Trees: Methods and Applications: In Honor of Boris Mirkin's 70th Birthday
And −ρ are increasing; 11. −ϕ and ρ are convex. 8. −ϕ and −ρ are convex; Fig. 3 Description of the polynomial solvable classes for the minimization problem 1. 2. 3. 4. 5. (ϕ1 ◦ v1 )(x) > 0 for x ∈ Bn ; it is possible to evaluate the objective function in polynomial time; v1 and v2 take integral values on Bn ; v1 and v2 are supermodular; one of the following conditions is satisfied: • v1 or v2 takes a polynomial number of distinct values on Bn ; • v1 and v2 are both linear; • v1 or v2 is monotone and the application x → (v1 (x), v2 (x)) is weakly bijective; 6.
The next result will be used to establish sufficient conditions on the functions ' and for the set X C to contain an optimal solution of problem (CFP). Proposition 4. x k /: Proof. x/ Á 28 P. Hansen and C. Meyer where we used (14). Since each term delimited by a pair of parentheses in the denominators is strictly positive, a simple manipulation gives the announced inequality. We terminate by pointing out that another characterization of the breakpoints can be found in Gallo and Simeone , as well as a different approach to compute the breakpoint vertex set that requires to solve a constrained version of problem PARAM.
Since x 00 and y 00 are comparable, the weak bijection property implies x 00 D y 00 . Since y 00 Ä x 0 , we conclude that SFM. / has the Strong Decreasing Optimal Solution Property. A Polynomial Algorithm for a Class of 0–1 Fractional Programming Problems. . Assume now that f is increasing. x / : (7) The left-hand side of (7) is nonpositive by submodularity of g while the right-hand side is nonnegative since 10 > 100 > 0 and since f is supermodular and increasing. Therefore all inequalities must hold at equality, in particular inequalities (4)–(5).