A Mathematical View of Interior-Point Methods in Convex Optimization
SIAM, 2001 - 117 strani
This compact book, through the simplifying perspective it presents, will take a reader who knows little of interior-point methods to within sight of the research frontier, developing key ideas that were over a decade in the making by numerous interior-point method researchers. It aims at developing a thorough understanding of the most general theory for interior-point methods, a class of algorithms for convex optimization problems. The study of these algorithms has dominated the continuous optimization literature for nearly 15 years. In that time, the theory has matured tremendously, but much of the literature is difficult to understand, even for specialists. By focusing only on essential elements of the theory and emphasizing the underlying geometry, A Mathematical View of Interior-Point Methods in Convex Optimization makes the theory accessible to a wide audience, allowing them to quickly develop a fundamental understanding of the material.
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a-val affine spaces algorithm analytic center approximation Assume f e asymptotically feasible barrier method Bx(x central path conjugate functional Consequently convex functional CONVEX OPTIMIZATION convex set Corollary decomposition define definition of self-concordance denote differentiable domain dot product dual feasible dual instance duality theory eigenvalues equivalent f e SC Farkas lemma feasible points following theorem Frobenius norm gw(x gx(x Hence Hessian implies inequality interior-point methods intrinsically self-conjugate ipm theory iterations line searches linear operator logarithmic barrier function minimizer Nesterov and Nemirovskii Nesterov-Todd directions Newton step Newton's method nonnegative orthant nullspace optimization problem orthogonal pd matrices primal and dual primal instance Proposition prove reference inner product rely s.t. Ax satisfying scaling point self-adjoint self-concordant functional self-scaled cone strong duality strongly nondegenerate self-concordant subspace suffices to show symmetric trace product univariate functional x e Df