A Study of Indicators for Identifying Zero Variables in Interior-Point Methods

Abstract

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Using Indicators in Finite Termination Procedures

“…Specifically, they defined (12) Mehrotra ancl Ye [21] proved that when 23~was defined as in (12) the optimal partition could be identified in iinite time for algorithms that generate iteration sequences that satisfy centrality measure (6). For numerical experiments, the authors used the Tapia indicators and variables as indicators in tandem.…”

“…In this case, the solution set S has the following structure (see E1-Bakry, Tapia, and Zhang [6] for a proof): (i) all points in the relative interior satisfy strict complementarily; (ii) the zero-nonzero pattern of points in the relative interior is invariant. For any (Z", g",~") in the relative interior of the solution set of (2), we deilne the index sets~and N as ={j:ti~> 0,1<j<2n)andfi= {j:i~=0,1<j<2n}.…”

“…The term indicator denotes a function that identifies active constraints at the solution of a constrained optimization problem, see Tapia [27] and E1-Bakry [5]. Commonly used indicators include variables as indicators, the primal-dual indicator, and the Tapia indicator.…”

The spectral bundle method with second-order information

“…Summarizing, the new candidate solution y + is determined through the bundle P and the scaling matrix H t which mimicks the second-order term H(Ũ) in step OW4; compare in particular (11) and (5). A quadratic SDP of the form (7) has to be solved.…”

Q-superlinear convergence of the iterates in primal-dual interior-point methods

“…Moreover all the nonzero entries of x * have the corresponding indices in B and all the nonzero entries of s * have the corresponding indices in N (cf. [7], [6]). Therefore any solution satisfying (29) is called a maximal solution.…”