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Extra info for Advanced Computational Methods for Knowledge Engineering: Proceedings of the 2nd International Conference on Computer Science, Applied Mathematics and Applications (ICCSAMA 2014)
Convex analysis approach to DC programming: Theory, Algorithms and Applications. Acta Mathematica Vietnamica 22(1), 289– 355 (1997) 3. : A DC Optimization algorithm for solving the trust region subproblem. SIAM J. Optim. 8(2), 476–505 (1998) 4. : Solving a class of linearly constrained indeﬁnite quadratic problems by DC Algorithms. Journal of Global Optimization 11, 253– 285 (1997) 5. : DC Programming: Theory, Algorithms and Applications. The State of the Art (28 pages). In: Proceedings of the First International Workshop on Global Constrained Optimization and Constraint Satisfaction (Cocos 2002), Valbonne-Sophia Antipolis, France, October 2-4 (2002) 6.
Huynh, and T. Pham Dinh with gk (x) := gk (x) + βk p1 (x); (23) hk (x) := h0 (x) + βk p2 (x). (24) By Assumption 2, one has ρk := ρ(gk ) + ρ(hk ) ≥ ρ(g0 ) + ρ(h0 ) + βk (ρ(p1 ) + ρ(p2 )) > 0. ; since y k ∈ ∂hk (xk ), then y k , xk+1 − xk ≤ hk (xk+1 ) − hk (xk ) − ρ(hk ) k x − xk+1 2 2 . (25) . (26) On the other hand, since xk+1 ∈ ∂(g + χC )∗ (y k ), then y k , xk − xk+1 ≤ gk (xk ) − gk (xk+1 ) − ρ(gk ) k x − xk+1 2 2 By inequalities (25) and (26), one obtains ϕk (xk ) − ϕk (xk+1 ) ≥ ρk k x − xk+1 2 2 .
Dinh xi := xi y(i−m) i=1,. . ,n i=n+1,. . ,N and x¯i := x ¯i y¯(i−m) i=1,. . ,n i=n+1,. . ,N. The proof of the proposition 1 is obvious. This result demonstrates that QMIFP and BMIFP are equivalent in feasibility, and their solution sets. Obviously, solving such a QMIFP is also a NP-hard problem, but QMIFP is easy to be reformulated as a DC program which could be solved eﬃciently with our proposed methods. In next section, we will focus on the DC reformulation of QMIFP. 3 DC Reformulation of QMIFP We know that QMIFP (2) is equivalent to a concave minimization problem over compact convex set consists of LMIs and bounds in the variables (see ).