(. ()Tj 0 Tc /F2 1 Tf 34 0 obj 0.5798 0 TD (I)Tj 0.333 Tc /F2 1 Tf /F5 1 Tf /F3 1 Tf [(],)-301.7(Chapter)-301.8(8,)-301.7(Section)-301.8(8)0(.2. (S)Tj /F1 1 Tf (i)Tj 0.4503 Tc (f)Tj 0.0001 Tc 0.3541 0 TD (i)Tj )-467.2(When)]TJ ()Tj 0.0001 Tc 0.1667 Tc /F3 1 Tf (S)Tj 0.9448 0 TD (. /F4 7 0 R 20.6626 0 0 20.6626 120.879 590.4661 Tm [(c)50.1(onvex)]TJ 6.5822 0 TD [(,)-546.8(for)-507.4(any)-507.7(se)50.1(quenc)50.1(e)-508.3(of)]TJ 11.9551 0 0 11.9551 72 736.329 Tm 20.6626 0 0 20.6626 140.004 436.3051 Tm /F4 1 Tf (\))Tj (S)Tj [(G)361.6(i)361.5(v)387.6(e)361.5(na)361.4(na)361.4()361.7(n)361.4(es)361.5(p)361.4(a)361.4(c)361.5(e)]TJ 20.6626 0 0 20.6626 232.173 292.4041 Tm (Two are shown, drawn in green and blue). 14.3462 0 0 14.3462 338.004 254.973 Tm [(con)26.1(v)-13(\()]TJ /F3 6 0 R 14.3462 0 0 14.3462 119.646 433.2001 Tm /F4 1 Tf -20.6834 -1.2057 TD 0 -1.2057 TD /F3 1 Tf 0.0041 Tc 0.0001 Tc 1.9361 0 TD 0.3338 0 TD (. (,)Tj [(of)-400.3(p)-26.2(o)-0.1(in)26(ts)-399.9(in)]TJ /F4 1 Tf 0.6991 0 TD 0.6904 0 TD (b)Tj (,)Tj 0 Tc 0.7836 0 TD /F7 1 Tf )-761.6(BASIC)-326.4(P)0(R)27.3(O)-0.3(PER)81.5(TIES)-326.3(OF)-326.1(CONVEX)-326.7(SETS)]TJ /F3 1 Tf 2.262 0 TD 0.0001 Tc (\()Tj (v)Tj /F4 1 Tf (C)Tj (i)Tj 1.2113 0.95 TD (S)Tj ()Tj /F5 1 Tf x��TKs1��أ����8�. 31.1377 0 TD It can be proved that under mild conditions midpoint convexity implies convexity. /F2 1 Tf 0.9861 0 TD 0 Tc 11.9551 0 0 11.9551 72 736.329 Tm 0.0001 Tc /F4 1 Tf ()Tj 0.0001 Tc /F2 1 Tf /F4 1 Tf 0.6608 0 TD (m)Tj /F4 1 Tf /F2 1 Tf /F4 1 Tf 0.3809 0 TD /F5 1 Tf [(short,)-301.8(requires)-301.9(a)-301.9(l)0(ot)-301.9(of)-301.8(creativit)26.1(y)78.3(. 0.2781 Tc /F2 5 0 R /F2 1 Tf /Length 2115 (E)Tj [(EODOR)81.5(Y)0(�S)-326.3(THEOREM)]TJ 0.0001 Tc /F4 1 Tf /F2 1 Tf 0 Tc 0.9975 0 TD >> -0.0001 Tc (f)Tj /F5 8 0 R 0.9974 0.7501 TD /F4 1 Tf [(of)-350.2(c)50.2(onvex)]TJ 6.0843 0 TD 0.9443 0 TD 0.8163 0 TD (|)Tj 9.9797 0 TD (|)Tj /F4 1 Tf ()Tj /F2 1 Tf [(\(1\))-402.4(i)0.1(s)-402.8(i)0.1(t)-402.4(p)-26.1(ossible)-402.4(ha)26.2(v)26.2(e)-402.4(a)-402.4(�)0.2(xed)-402.9(b)-26.1(ound)-402(on)-402.5(the)-402.8(n)26.1(um)26(b)-26.1(e)0.1(r)-401.9(o)0(f)]TJ 0.0001 Tc 0 g -6.969 -1.2052 TD 0.5001 0 TD 1.2209 0 TD [(of)-388(a)-388.1(nonempt)26.2(y)-387.6(con-)]TJ (E)Tj /F3 1 Tf -15.875 -1.2052 TD /F5 1 Tf (and)Tj 0 Tc (S,)Tj [(Given)-429.6(an)-429.2(ane)-429.4(sp)50(ac)50.1(e)]TJ 0.8564 0 TD 20.6626 0 0 20.6626 278.838 258.078 Tm /F4 1 Tf /F5 1 Tf 8.3171 0 TD 1.4579 0 TD Formally, if X:= fx i 2Rn j1 i mgis an arbitrary set of points, then its convex hull is the set obtained by taking all possible convex combinations of the points in X. 0.849 0 TD /F3 1 Tf /F4 1 Tf 1.0606 0 TD /F2 5 0 R /F2 1 Tf (})Tj Intuitively, if we think of R2 or R3, a convex set of vectors is a set that contains all the points of any line segment joining two points of the set (see the next gure). 0.3509 Tc 14.3462 0 0 14.3462 155.538 573.402 Tm (0)Tj /F4 1 Tf /F2 1 Tf [(Observ)26.2(e)-398.9(t)0.1(hat)-398.9(if)]TJ ()Tj /F2 1 Tf /F2 1 Tf 0.5893 0 TD (S)Tj 0.6608 0 TD 0.5893 0 TD /F4 1 Tf /F2 1 Tf (a)Tj (j)Tj [(\)i)283.7(st)283.6(h)283.5(e)]TJ 20.6626 0 0 20.6626 333.045 663.519 Tm /F2 1 Tf The second development is the discovery that convex optimization problems (beyond least-squares and linear programs) are more prevalent in practice than was previously thought. /F7 1 Tf 10.0333 0 TD /F2 1 Tf 0.6608 0 TD /F5 8 0 R 0.8564 0 TD 12.9565 0 TD [(ve)-26.1(x)-26.2(. (S)Tj 0 0 1 rg /F4 1 Tf (m)Tj 14.3462 0 0 14.3462 440.334 265.683 Tm (\)=)Tj 42 0 obj 0.0001 Tc endobj (a)Tj 4.0627 0 TD /F4 1 Tf (b,)Tj (f)Tj Conv(S) ∨ Conv(T) = Conv(S ∪ T) = Conv(Conv(S) ∪ Conv(T)).The intersection of any collection of convex sets is itself convex, so the convex subsets of a (real or complex) vector space form a complete lattice. 14.3462 0 0 14.3462 369.252 261.6151 Tm ()Tj 5.0201 0 TD [(p)-26.2(o)-0.1(in)26(ts,)-456.4(or)-425.1(is)-425.6(it)-425.6(p)-26.2(o)-0.1(ssible)-425.6(to)-425.6(only)-425.2(c)0(onsider)-425.6(a)-425.6(s)0(ubset)-425.1(with)]TJ 14.3462 0 0 14.3462 311.571 191.9641 Tm (+1)Tj /F2 1 Tf ()Tj 0.0001 Tc /F4 1 Tf (S)Tj 1.3559 0 TD /F2 1 Tf /F4 1 Tf 226.093 654.17 l 1.1068 0 TD [(a,)-166.6(b)]TJ 7.3645 0 TD /F1 1 Tf 14.3462 0 0 14.3462 377.244 490.701 Tm endobj /F2 1 Tf (\))Tj 2.1366 0 TD 38.1668 0 TD 0.9539 0 TD 0 Tc (de�ning)Tj (S)Tj (=1)Tj /F5 1 Tf /F5 1 Tf /F3 1 Tf . (+)Tj 0.3541 0 TD 0.5893 0 TD 391.038 705.193 l 0.333 Tc << 2.8875 0 TD [(\))-350(i)0(s)-350(t)0.2(he)-349.6(c)50.2(onvex)-350.1(hul)-50(l)-350.1(of)]TJ 20.6626 0 0 20.6626 199.062 590.4661 Tm (i)Tj (\))Tj 14.3462 0 0 14.3462 190.152 289.299 Tm [(,)-273.5(d)-0.1(enoted)-266.2(cone\()]TJ 0.5558 0 TD 357.557 597.477 l /Length 5929 ()Tj 0.3541 0 TD (a)Tj (b)Tj 14.3462 0 0 14.3462 448.479 623.217 Tm (q)Tj 1.386 0 TD /F6 9 0 R 0 Tc (I)Tj (+)Tj /F2 1 Tf [(a)-353.6(h)26.1(yp)-26.1(erplane)]TJ 13 0 obj (1\()Tj 0.0001 Tc 20.6626 0 0 20.6626 72 702.183 Tm )-813.2(In)-437.4(case)-437.3(1,)-471.2(assuming)]TJ 357.557 597.477 m 0.7836 0 TD ([)Tj 43 0 obj [(tices)-301.9(b)-26.2(elong)-301.9(to)]TJ /F2 1 Tf (a)Tj 0.3391 Tc 0 Tc (\()Tj [(,)-306.6(d)-0.1(enoted)-305.9(b)26(y)-305.4(dim)]TJ 6.1156 0 TD /F2 1 Tf 0.0001 Tc /F2 1 Tf /F9 1 Tf /F3 1 Tf By applying this property several times, we observe that a convex set Cmust contain any convex … [(W)78.6(e)-290.6(get)-290.5(t)0(he)-290.1(feeling)-290.6(t)0(hat)-290.5(triangulations)-290.1(pla)26.1(y)-290.6(a)-290.1(crucial)-290.5(r)0(ole,)]TJ /F5 1 Tf [(tan)26(t)-299.2(role)-299.3(in)-299.3(con)26(v)26.1(ex)-299.3(optimization. 3.3671 0 TD /F5 1 Tf /F2 1 Tf 0 Tw /F5 1 Tf /F4 1 Tf (3)Tj /ProcSet [/PDF /Text ] (a)Tj 2.0442 0 TD 0.9443 0 TD ({)Tj [(CHAPTER)-327.3(3. (\))Tj >> (c)Tj 3.175 0 TD )Tj 0 Tc [(,i)366.7(f)]TJ 379.485 628.847 m 0 -1.2052 TD (i)Tj 24.7871 0 0 24.7871 72 624.873 Tm 1.0554 0 TD (S)Tj 3.3313 0 TD /F3 1 Tf 0.8359 0 TD 0.9857 0 TD /F2 1 Tf (H)Tj /F4 1 Tf 14.3462 0 0 14.3462 86.922 561.234 Tm [(,)-448.7(for)]TJ 9.6003 0 TD /GS1 11 0 R 0.1237 -0.7932 TD -0.0003 Tc 1.0559 0 TD 20.6626 0 0 20.6626 94.833 242.5891 Tm 0.3337 0 TD /F6 1 Tf 0.0527 -0.7187 TD 0.5893 0 TD [(is)-250.2(any)-250.1(c)50.2(o)0(mp)50.1(act)-250.3(subset)-251.2(o)0(f)]TJ @m1�%I�Ƙ[�ǝD 1.1604 0 TD [(con)26.1(v)-12.6(\()]TJ 0 Tc (q)Tj stream 226.093 597.477 m /F4 1 Tf (\()Tj /F5 1 Tf /F4 1 Tf /F4 1 Tf /F5 1 Tf [(Bounded)-263.2(c)0(on)26(v)26.1(e)0(x)-263.2(sets)-263.5(arising)-263.6(a)-0.1(s)-263.1(t)0(he)-263.6(in)26(tersection)-263.2(o)-0.1(f)-263.5(a)-263.6(�nite)]TJ -14.9132 -1.2052 TD 0.6669 0 TD ()Tj 3.9573 0 TD endstream (98)Tj 0 0 1 rg -13.8787 -1.2052 TD 0.6669 0 TD 0.0001 Tc 0.9274 0 TD 1.1255 0 TD /F3 1 Tf /F5 1 Tf 0.2989 Tc /F4 1 Tf 1.4971 0 TD [(�nite)-366.3(set)-365.9(of)-366.3(cardinalit)26.1(y)]TJ 226.093 654.17 m /F2 1 Tf [(\). (S)Tj 0 Tc ()Tj /F2 1 Tf /F5 1 Tf endobj 0.0001 Tc /F4 1 Tf /F4 1 Tf (with)Tj 15 0 obj -18.5395 -1.2052 TD 0.0001 Tc (\)=)Tj /F2 1 Tf (+)Tj -18.0694 -1.2052 TD /F5 1 Tf /F8 16 0 R /F7 1 Tf 13.4618 0 TD 0.2781 Tc 0.3541 0 TD 0.2781 Tc (I)Tj 11.9551 0 0 11.9551 72 736.329 Tm 0.6669 0 TD /F2 1 Tf 0.3337 0 TD /F4 1 Tf 2.3979 0 TD 0 -1.2052 TD 2.4118 0 TD >> ()Tj [(,s)315.1(p)365(a)314.9(n)314.8(n)314.9(e)365.1(d)8.3(b)315(y)]TJ >> /F3 1 Tf 0.585 0 TD ()Tj -0.1302 -0.2529 TD /F3 1 Tf /F4 1 Tf (\()Tj 1.4562 0 TD 14.3462 0 0 14.3462 253.656 264.3961 Tm )]TJ 0 Tc endstream /F7 1 Tf /F5 1 Tf /F4 1 Tf 0 Tc /F9 1 Tf [(is)-202.5(of)-202.5(course)-202.1(con)26(v)26.1(ex. 20.6626 0 0 20.6626 137.988 493.7971 Tm 0.0041 Tc ET /F3 6 0 R )Tj /F2 1 Tf /F5 1 Tf (and)Tj /F4 1 Tf /F5 1 Tf 11.7021 0 TD (H)Tj 0 Tw (\()Tj A convex set is a set which contains all of the line segments whose endpoints are in it. 0.0001 Tc /F4 1 Tf /F4 1 Tf -15.5744 -1.2057 TD 0 Tc 20.6626 0 0 20.6626 501.255 518.709 Tm (and)Tj Then, given any (nonempty) subset S of E, there is a smallest convex set containing S denoted by C(S)(or conv(S)) and called the convex hull of S (namely, theintersection of all convex sets containing S).The aﬃne hull of a subset, S,ofE is the smallest aﬃne set contain- 20.6626 0 0 20.6626 295.929 258.078 Tm >> (for)Tj 14.3462 0 0 14.3462 389.178 649.272 Tm ()Tj (b)Tj 0.0001 Tc /F5 1 Tf 20.6626 0 0 20.6626 255.204 663.519 Tm (=0)Tj /Font << (a)Tj /F2 1 Tf /ProcSet [/PDF /Text ] 20.6626 0 0 20.6626 417.555 258.078 Tm 0.5001 0 TD /F4 1 Tf 0.3337 0 TD /F8 16 0 R (I)Tj )Tj /F5 1 Tf [(v)26.1(ertices)-301.9(b)-26.2(elong)-301.9(to)]TJ ()Tj (\()Tj 17.2155 0 0 17.2155 72 704.577 Tm /F4 1 Tf -0.0001 Tc (L)Tj /F4 1 Tf 1.7998 0 TD The convex-hull operation is needed for the set of convex sets to form a lattice, in which the "join" operation is the convex hull of the union of two convex sets . /F2 1 Tf 0 0 1 rg /F4 1 Tf 2.1361 0 TD /F2 1 Tf /F2 1 Tf (})Tj 20.6626 0 0 20.6626 445.671 344.3701 Tm -0.0003 Tc 0.0001 Tc [(c)50.2(o)0(mbinations)]TJ 7.3645 0 TD /F2 1 Tf /F8 16 0 R /F9 1 Tf /F7 1 Tf [(\(2\))-301.4(I)0(s)-301.4(i)0(t)-301(n)-0.1(ecessary)-301.5(to)-301(consider)-301.4(con)26(v)26.1(ex)-301.1(com)25.9(b)-0.1(inations)-301(of)-301.4(all)]TJ C of a convex set C, and its lineality space L C = R C \( R C): 0 x + ay x Convex Set C Recession Cone RC y For S k:convex, the set of asymptotic directions of fS kgis the set of nonzero d 2\ kR S k. The vector sum of a compact set and a polyhedral cone (e.g., a polyhedral set) is re-tractive. /F3 1 Tf ET /F2 1 Tf [(,)-315.4(t)0.2(hat)-306.9(is,)]TJ 0 Tc 2.4384 0 TD (E)Tj 1.0689 0 TD /F4 1 Tf 1.9745 0 TD /F4 1 Tf (c)Tj 0.1667 Tc 0.7671 0 TD [(0)-917.3(f)0.1(or)-301.8(all)]TJ 0.7919 0 TD BT /F2 1 Tf 1 0 0 RG /F2 1 Tf /F7 10 0 R 0.3999 0 TD endobj 0 Tc 220.959 620.154 m /F2 1 Tf 1.6295 0 TD 1.143 0 TD (i)Tj 0.6669 0 TD 11.9551 0 0 11.9551 300.15 74.6401 Tm (i)Tj endobj /F4 1 Tf ()Tj [(space)-301.8(o)0(f)]TJ 14.3462 0 0 14.3462 358.362 404.769 Tm (i)Tj )Tj 2.8204 0 TD /F2 1 Tf ��ΦP{p�������^�}b�'ځ��H��Dq��l��|���' ʉc��P�}^������NZ�~bS��2e��֬.��0f*�P>���6S�P���-���%�(cH����kMLl�r���5Pg������v��!�(E���+����r�%�2o4h��gj�N5�J����২6 AM��~$���/w��/b����4�za�!�AY�[�|�jm�J�Y,��
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