jointly constrained biconvex programming - Convex Optimization

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280FAIZ A. AL-KHYYAL AND JAMES E. FALKthe smallest integer larger than k, such that (xk'+,, y k+l) lies in the chosen set. In thisway, we can assume that the sequence ((x k, yk')E T is associated with a nestedsequence of sets {( k(t) }e T, i.e.,Clearly,Q = Qk( 1) D .k(2) D ..and(X,^) E n ^k(t,e Tfor each i.Moreover, we must have:andi, =' limTl ('t < xi < lim m L ( - ,m, -lim m (t) < yi < limTMik(t) ' Mt ETteTeither 4l = x or xj = L/either j =y, or = MIfor some I, or otherwise we could not have {(x k, ykt)} converging to (x, y).Now supposeandTheni
JOINTLY CONSTRAINED BICONVEX PROGRAMMING281For notational convenience, letHk (xi, Yi) = Vexulx/yi ,andhik (xi, Yi) = m,kjxi + l ijyi-lJ im,ik,so thath. (xii, Yi) = MikJxi + L,kiyi - LJMikj,n(x, Y) = H (xi yi) = 2 max { hA(xi, yi), h,(xi,i=Ini-Iwhen (x, y) E flki.Let yi be the solution value and (ai, /,) the solution point of the maximum normproblem max{ V xiyi[|: (xi, Yi) F i).}That is,7i = max( IIVxyill :(xi,,i) E 2i)= |1(i I,ai)||.Note that (at, /,i) must be a corner point of the set Qi. Furthermore, for every j andall k,m ax{| V -i,|hl ||ll, hi /II l i, i 1,..., n,since the refining rule ensures that 1i < lk, mi < mJ, Li > L,, and Mi > Miki.Consider the line Yi in fi going through an arbitrary point (xi, Yi) E Si in thedirection (ai, ,). Construct the line in ?3 having domain Yz and slope yi. Thesurface of revolution of this line about the vertical axis passing through the point(xi, y,xiyi) is a nonconvex cone. Truncate the cone, top and bottom, at Euclideandistances of xiyi- e/n and xiyi + e/n away from the xi, yi-plane. This yields thecone C of Figure 3.The projection of C onto the xi, yi-plane is the set {(ui,vi): II(xi, yi)- (ui,,vi)l< e/ny,}. Let y = maxyi and 8 = E/ny. Then we can write: given c > 0, for any(xi, yi),(ui, vi) E- i,for every i = 1, . . .,n.II(xi, y) - (ui,vi)II < 8=|lxi- uivil < E/nyi))c/n \ C /c/n(x.,yi, x.y.)(ai,3i)FIGURE 3. Cone generated by line in R 3 having domain -Zi and slope Yi,.
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jointly constrained biconvex programming - Convex Optimization

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