TY - JOUR
T1 - Doubly nonnegative relaxations are equivalent to completely positive reformulations of quadratic optimization problems with block-clique graph structures
AU - Kim, Sunyoung
AU - Kojima, Masakazu
AU - Toh, Kim Chuan
N1 - Funding Information:
The research was supported by NRF 2017-R1A2B2005119. This research was supported by Grant-in-Aid for Scientific Research (A) 26242027. This research is supported in part by the Ministry of Education of Singapore under Academic Research Fund Grant Number: R-146-000-257-112.
Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2020/7/1
Y1 - 2020/7/1
N2 - We study the equivalence among a nonconvex QOP, its CPP and DNN relaxations under the assumption that the aggregate and correlative sparsity of the data matrices of the CPP relaxation is represented by a block-clique graph G. By exploiting the correlative sparsity, we decompose the CPP relaxation problem into a clique-tree structured family of smaller subproblems. Each subproblem is associated with a node of a clique tree of G. The optimal value can be obtained by applying an algorithm that we propose for solving the subproblems recursively from leaf nodes to the root node of the clique-tree. We establish the equivalence between the QOP and its DNN relaxation from the equivalence between the reduced family of subproblems and their DNN relaxations by applying the known results on: (1) CPP and DNN reformulation of a class of QOPs with linear equality, complementarity and binary constraints in 3 nonnegative variables. (2) DNN reformulation of a class of quadratically constrained convex QOPs with any size. (3) DNN reformulation of LPs with any size. As a result, we show that a QOP whose subproblems are the QOPs mentioned in (1), (2) and (3) is equivalent to its DNN relaxation, if the subproblems form a clique-tree structured family induced from a block-clique graph.
AB - We study the equivalence among a nonconvex QOP, its CPP and DNN relaxations under the assumption that the aggregate and correlative sparsity of the data matrices of the CPP relaxation is represented by a block-clique graph G. By exploiting the correlative sparsity, we decompose the CPP relaxation problem into a clique-tree structured family of smaller subproblems. Each subproblem is associated with a node of a clique tree of G. The optimal value can be obtained by applying an algorithm that we propose for solving the subproblems recursively from leaf nodes to the root node of the clique-tree. We establish the equivalence between the QOP and its DNN relaxation from the equivalence between the reduced family of subproblems and their DNN relaxations by applying the known results on: (1) CPP and DNN reformulation of a class of QOPs with linear equality, complementarity and binary constraints in 3 nonnegative variables. (2) DNN reformulation of a class of quadratically constrained convex QOPs with any size. (3) DNN reformulation of LPs with any size. As a result, we show that a QOP whose subproblems are the QOPs mentioned in (1), (2) and (3) is equivalent to its DNN relaxation, if the subproblems form a clique-tree structured family induced from a block-clique graph.
KW - Aggregate and correlative sparsity
KW - Block-clique graphs
KW - Completely positive and doubly nonnegative matrix completion
KW - Equivalence of doubly nonnegative relaxations and completely positive programs
KW - Exact optimal values of nonconvex QOPs
KW - Sparsity of completely positive reformulations
UR - http://www.scopus.com/inward/record.url?scp=85078865739&partnerID=8YFLogxK
U2 - 10.1007/s10898-020-00879-y
DO - 10.1007/s10898-020-00879-y
M3 - Article
AN - SCOPUS:85078865739
VL - 77
SP - 513
EP - 541
JO - Journal of Global Optimization
JF - Journal of Global Optimization
SN - 0925-5001
IS - 3
ER -