By Zdenek Dostál
Quadratic programming (QP) is one complicated mathematical approach that enables for the optimization of a quadratic functionality in different variables within the presence of linear constraints. This booklet offers lately built algorithms for fixing huge QP difficulties and specializes in algorithms that are, in a feeling optimum, i.e., they could resolve very important periods of difficulties at a price proportional to the variety of unknowns. for every set of rules offered, the publication information its classical predecessor, describes its drawbacks, introduces variations that increase its functionality, and demonstrates those advancements via numerical experiments. This self-contained monograph can function an introductory textual content on quadratic programming for graduate scholars and researchers. also, because the answer of many nonlinear difficulties should be diminished to the answer of a series of QP difficulties, it will possibly even be used as a handy creation to nonlinear programming.
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Additional resources for Optimal Quadratic Programming Algorithms: With Applications to Variational Inequalities (Springer Optimization and Its Applications)
Example text
7. Let A ∈ Rn×n denote a symmetric matrix, let B ∈ Rm×n denote a matrix of the rank r, 0 < r ≤ m < n, and > 0. Then λr (BT B) > 0 and λr (A ) ≥ λn (A) + λr (BT B), λr+1 (A ) ≤ λ1 (A). 51) 26 1 Linear Algebra Proof. 22), we can find an orthogonal matrix U such that UT BT BU = diag(γ1 , . . , γn ) with γi = λi (BT B) and γ1 ≥ · · · ≥ γr > γr+1 = γn = 0. Thus UT (A + BT B)U = UT AU + UT BT BU = E+ G F FT H with G = diag(γ1 , . . , γr ) and UT AU = E F , FT H where H is a square matrix of the order n − r.
2. Let A ∈ Rn×n be a symmetric positive semidefinite matrix, let B ∈ Rm×n , > 0, and let KerA ∩ KerB = {o}. Then A is positive definite. Proof. If x = o and KerA ∩ KerB = {o}, then either Ax = o or Bx = o. 27) equivalent to A1/2 x = o, we get for > 0 xT A x = xT Ax + Bx 2 = A1/2 x 2 + Bx 2 > 0. Thus A is positive definite. 2, using KerA = {o}, that A is also positive definite. 6) to get A−1 = A−1 − A−1 BT ( −1 I + BA−1 BT )−1 BA−1 . 39) The following lemma shows that A can be positive definite even when A is indefinite.
8, we get that x ∈ ΩS is the minimizer of a convex quadratic function f on ΩS if and only if ∇f (x) is orthogonal to S. 18). 18). Its second component λ is called a vector of Lagrange multipliers or simply a multiplier. We shall often use the notation x or λ to denote the components of a KKT pair that are uniquely determined. 8 has a simple geometrical interpretation. 24) requires that the gradient of f at a solution x is orthogonal to KerB, the set of feasible directions of ΩE , so that there is no feasible decrease direction as illustrated in Fig.