Splines are low-order polynomials that are used to interpolate subsets of the data. The higher-order polynomial captures all of the variation, and in situations such as medical imaging and graphics, where smooth surfaces are desirable, small variations detract from the presentation. Sometimes it is preferable to fit a lower order polynomial to the n+1 data. Splines are polynomial approximations where some data points are replaced by constraints on data and the interpolating polynomials. In fact, some of the n+1 data points can be replaced by other conditions or constraints, so that fewer than n+1 data points are needed. The Lagrange and Newton divided difference interpolating polynomials are based on the principle that n+1 points of data are needed to approximate a function of order n. MOGHE, in Numerical Methods in Biomedical Engineering, 2006 9.6.3 Splines This concept of parallel subspace property can also be extended to higher dimensions. Thus, in the case of a quadratic polynomial, four unidirectional searches will find the minimum point and in higher-order polynomials, more than four unidirectional searches may be necessary. Then, the point x ( 2 ) is obtained by performing a unidirectional search along ( 0, 1 ) T from y ( 1 ), and finally, the point y ( 2 ) is found by a unidirectional search along the direction ( 1, 0 ) T from the point x ( 2 ).įor a quadratic polynomial, the minimum lies in the direction ( y ( 2 ) − y ( 1 ) ), but for higher-order polynomials, the true minimum may not lie in the aforementioned direction. The point y ( 1 ) is obtained by performing a unidirectional search along ( 1, 0 ) T from the point x ( 1 ). Instead of using two points x ( 1 ) and x ( 2 ) and a direction vector S to create one pair of conjugate directions, one point x ( 1 ) and both coordinate directions ( 1, 0 ) T and ( 0, 1 ) T can be used to create a pair of conjugate directions d and ( y ( 2 ) − y ( 1 ) ). The vector ( y ( 2 ) − y ( 1 ) ) forms a conjugate direction with the original direction vector S. For quadratic functions, we can say that the minimum of the function lies on the line joining the points y ( 1 ) and y ( 2 ). Thus, if two arbitrary points x ( 1 ) and x ( 2 ) and an arbitrary search direction S are chosen, two unidirectional searches, one from each point will create two points y ( 1 ) and y ( 2 ). Then the direction ( y ( 2 ) − y ( 1 ) ) is conjugate to S or, in other words, the quantity ( y ( 2 ) − y ( 1 ) ) T CS is zero. Putinar (1993) gave a useful characterization of positive polynomials on certain sets described by polynomial inequalities, and it provides the basis for Lasserre and Thanh's method. Hilbert showed in 1888 that this is not true in general, but the first counterexample was given by Motzkin in 1967. All nonnegative polynomials in one variable as well as all nonnegative polynomials in two variables of degree ≤ 4 can be represented as a sum of squares of polynomials. That this is the case was proven in 1925 by Emil Artin Roy (1999). Hilbert's Seventeenth Problem asked whether all nonnegative polynomials on R n can be represented as sums of squares of rational functions. Lasserre and Thanh (2012) described an underestimation method relying on results from algebraic geometry. One of these methods, though not very promising for general functions, has good theoretical properties on quadratic functions. A number of new algorithms for determining nondiagonal perturbation parameters are described and compared in Skjäl and Westerlund (2012). The latter paper contains a review of the origins and variants of αBB methods. (2004) and rigorously studied by Skjäl et al. The extension of the αBB methodology to “nondiagonal” perturbations was suggested by Akrotirianakis et al. One advantage of convex underestimation approaches is that they can be combined to handle more general constraints and objectives. A “definitive” algorithm for these problems seems unlikely in light of the NP-hardness, rather, different approaches will dominate in performance on different problem subclasses. See Floudas and Visweswaran (1995) for a review of methods for quadratic programs (QP) and quadratically constrained quadratic programs (QCQP). Applications are found in, e.g., scheduling, facility allocation, supply chain and pooling problems. On the other hand quadratic expressions appear naturally and frequently in mathematical programming models. The hardships of NP-completeness are present already when minimizing quadratic functions with a single negative eigenvalue Pardalos and Vavasis (1991). Nondefinite quadratic functions are in a way the basic example of nonconvexity. 23rd European Symposium on Computer Aided Process EngineeringĪnders Skjäl, Tapio Westerlund, in Computer Aided Chemical Engineering, 2013 1 Introduction
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