We propose a method for support vector machine classification using indefinite kernels. Instead of directly minimizing or stabilizing a nonconvex loss function, our algorithm simultaneously computes support vectors and a proxy kernel matrix used in forming the loss. This can be interpreted as a penalized kernel learning problem where indefinite kernel matrices are treated as noisy observations of a true Mercer kernel. Our formulation keeps the problem convex and relatively large problems can be solved efficiently using the projected gradient or analytic center cutting plane methods. We compare the performance of our technique with other methods on several standard data sets.

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Local search with k-exchange neighborhoods, k-opt, is the most widely used heuristic method for the traveling salesman problem (TSP). This paper presents an effective implementation of k-opt in LKH-2, a variant of the Lin–Kernighan TSP heuristic. The effectiveness of the implementation is demonstrated with experiments on Euclidean instances ranging from 10,000 to 10,000,000 cities. The runtime of the method increases almost linearly with the problem size. LKH-2 is free of charge for academic and non-commercial use and can be downloaded in source code.

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Lift-and-project cuts for mixed integer programs (MIP), derived from a disjunction on an integer-constrained fractional variable, were originally (Balas et al. in Math program 58:295–324, 1993) generated by solving a higher-dimensional cut generating linear program (CGLP). Later, a correspondence established (Balas and Perregaard in Math program 94:221–245, 2003) between basic feasible solutions to the CGLP and basic (not necessarily feasible) solutions to the linear programming relaxation LP of the MIP, has made it possible to mimic the process of solving the CGLP through certain pivots in the LP tableau guaranteed to improve the CGLP objective function. This has also led to an alternative interpretation of lift-and-project (L&P) cuts, as mixed integer Gomory cuts from various (in general neither primal nor dual feasible) LP tableaus, guaranteed to be stronger than the one from the optimal tableau. In this paper we analyze the relationship between a pivot in the LP tableau and the (unique) corresponding block pivot (sequence of pivots) in the CGLP tableau. Namely, we show how a single pivot in the LP defines a sequence (potentially as long as the number of variables) of pivots in the CGLP, and we identify this sequence. Also, we give a new procedure for finding in a given LP tableau a pivot that produces the maximum improvement in the CGLP objective (which measures the amount of violation of the resulting cut by the current LP solution). Further, we introduce a procedure called
iterative disjunctive modularization. In the standard procedure, pivoting in the LP tableau optimizes the multipliers with which the inequalities on each side of the disjunction are weighted in the resulting cut. Once this solution has been obtained, a strengthening step is applied that uses the integrality constraints (if any) on the variables on each side of the disjunction to improve the cut coefficients by choosing optimal values for the elements of a certain monoid. Iterative disjunctive modularization is a procedure for approximating the simultaneous optimization of both the continuous multipliers and the integer elements of the monoid. All this is discussed in the context of a CGLP with a more general normalization constraint than the standard one used in (Balas and Perregaard in Math program 94:221–245, 2003), and the expressions that describe the above mentioned correspondence are accordingly generalized. Finally, we summarize our extensive computational experience with the above procedures.

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Finding a feasible solution of a given mixed-integer programming (MIP) model is a very important NP-complete problem that can be extremely hard in practice. Feasibility Pump (FP) is a heuristic scheme for finding a feasible solution to general MIPs that can be viewed as a clever way to round a sequence of fractional solutions of the LP relaxation, until a feasible one is eventually found. In this paper we study the effect of replacing the original rounding function (which is fast and simple, but somehow blind) with more clever rounding heuristics. In particular, we investigate the use of a diving-like procedure based on rounding and constraint propagation—a basic tool in Constraint Programming. Extensive computational results on binary and general integer MIPs from the literature show that the new approach produces a substantial improvement of the FP success rate, without slowing-down the method and with a significantly better quality of the feasible solutions found.

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