## Taiwanese Journal of Mathematics

### Extension of Eaves Theorem for Determining the Boundedness of Convex Quadratic Programming Problems

#### Abstract

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 $l_p$ 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 constraints. 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.

#### Article information

Source
Taiwanese J. Math., Advance publication (2020), 13 pages.

Dates
First available in Project Euclid: 7 May 2020

https://projecteuclid.org/euclid.twjm/1588838410

Digital Object Identifier
doi:10.11650/tjm/200501

#### Citation

Nguyen, Huu-Quang; Nguyen, Van-Bong; Sheu, Ruey-Lin. Extension of Eaves Theorem for Determining the Boundedness of Convex Quadratic Programming Problems. Taiwanese J. Math., advance publication, 7 May 2020. doi:10.11650/tjm/200501. https://projecteuclid.org/euclid.twjm/1588838410

#### References

• R. J. Caron and W. Obuchowska, Unboundedness of a convex quadratic function subject to concave and convex quadratic constraints, European J. Oper. Res. 63 (1992), no. 1, 114–123.
• ––––, An algorithm to determine boundedness of quadratically constrained convex quadratic programmes, European J. Oper. Res. 80 (1995), no. 2, 431–438.
• V. V. Dong and N. N. Tam, On the solution existence of convex quadratic programming problems in Hilbert spaces, Taiwanese J. Math. 20 (2016), no. 6, 1417–1436.
• Z. Dostál, On solvability of convex noncoercive quadratic programming problems, J. Optim. Theory Appl. 143 (2009), no. 2, 413–416.
• B. C. Eaves, On quadratic programming, Management Sci. 17 (1971), 698–711.
• D. S. Kim, N. N. Tam and N. D. Yen, Solution existence and stability of quadratically constrained convex quadratic programs, Optim. Lett. 6 (2012), no. 2, 363–373.
• Z.-Q. Luo and S. Zhang, On extensions of the Frank-Wolfe theorems, Comput. Optim. Appl. 13 (1999), no. 1-3, 87–110.
• K. G. Murty, Linear Complementarity, Linear and Nonlinear Programming, Sigma Series in Applied Mathematics 3, Heldermann Verlag, Berlin, 1988.
• E. L. Peterson and J. G. Ecker, Geometric programming: duality in quadratic programming and $l_p$-approximation I, Proceedings of the Princeton Symposium on Mathematical Programming (Princeton Univ., 1967), 445–480, Princeton University Press, Princeton, N.J., 1970.
• ––––, Geometric programming: duality in quadratic programming and $l_p$-approximation II (Canonical programs), SIAM J. Appl. Math. 17 (1969), 317–340.
• ––––, Geometric programming: duality in quadratic programming and $l_p$-approximation III: Degenerate programs, J. Math. Anal. Appl. 29 (1970), 365–383.
• R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series 28, Princeton University Press, Princeton, N.J., 1970.
• T. Terlaky, On $l_p$ programming, European J. Oper. Res. 22 (1985), no. 1, 70–100.