TY - JOUR

T1 - Extension of eaves theorem for determining the boundedness of convex quadratic programming problems

AU - Nguyen, Huu Quang

AU - Nguyen, Van Bong

AU - Sheu, Ruey Lin

N1 - Funding Information:
This work was supported by Taiwan Ministry of Science and Technology under grants MOST 105-2115-M-006-005-MY2; MOST 106-2811-M-006-021 and MOST 107-2811-M-006-009.
Funding Information:
Received July 24, 2019; Accepted May 3, 2020. Communicated by Jein-Shan Chen. 2010 Mathematics Subject Classification. 90C20, 90C22, 90C32. Key words and phrases. convex quadratic programming, lp programming, Eaves theorem, recession cone. This work was supported by Taiwan Ministry of Science and Technology under grants MOST 105-2115-M-006-005-MY2; MOST 106-2811-M-006 -021 and MOST 107-2811-M-006-009. *Corresponding author.
Publisher Copyright:
© 2020, Mathematical Society of the Rep. of China. All rights reserved.

PY - 2020/12

Y1 - 2020/12

N2 - It is known that the boundedness of a convex quadratic function over a convex quadratic constraint (c-QP) can be determined by algorithms. In 1985, Terlaky transformed the said boundedness problem into an lp programming problem and then apply linear programming, while Caron and Obuchowska in 1995 proposed another iterative procedure that checks, repeatedly, the existence of the implicit equality con-straints. Theoretical characterization about the boundedness of (c-QP), however, does not have a complete result so far, except for Eaves’ theorem, first by Eaves and later by Dostál, which answered the boundedness question only partially for a polyhedral-type of constraints. In this paper, Eaves’ theorem is generalized to answer, necessarily and sufficiently, when the general (c-QP) with a convex quadratic constraint (not just a polyhedron) can be bounded from below, with a new insight that it can only be unbounded within an affine subspace.

AB - It is known that the boundedness of a convex quadratic function over a convex quadratic constraint (c-QP) can be determined by algorithms. In 1985, Terlaky transformed the said boundedness problem into an lp programming problem and then apply linear programming, while Caron and Obuchowska in 1995 proposed another iterative procedure that checks, repeatedly, the existence of the implicit equality con-straints. Theoretical characterization about the boundedness of (c-QP), however, does not have a complete result so far, except for Eaves’ theorem, first by Eaves and later by Dostál, which answered the boundedness question only partially for a polyhedral-type of constraints. In this paper, Eaves’ theorem is generalized to answer, necessarily and sufficiently, when the general (c-QP) with a convex quadratic constraint (not just a polyhedron) can be bounded from below, with a new insight that it can only be unbounded within an affine subspace.

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U2 - 10.11650/tjm/200501

DO - 10.11650/tjm/200501

M3 - Article

AN - SCOPUS:85096393598

VL - 24

SP - 1551

EP - 1563

JO - Taiwanese Journal of Mathematics

JF - Taiwanese Journal of Mathematics

SN - 1027-5487

IS - 6

ER -