5.9912 0 TD /F5 1 Tf 2.2019 0 TD 3.1. [(short,)-301.8(requires)-301.9(a)-301.9(l)0(ot)-301.9(of)-301.8(creativit)26.1(y)78.3(. 0.9975 0 TD )Tj /F4 1 Tf /F3 1 Tf (i)Tj 1.0554 0 TD (�)Tj (. 0 Tc /F2 1 Tf 0.585 0 TD 20.6626 0 0 20.6626 72 659.208 Tm 11.7064 0 TD Title: convex_set.ppt Author: marie claude vergne Created Date: 7/21/2010 1:17:22 PM /F2 1 Tf /F2 1 Tf /F4 1 Tf 0.5893 0 TD 0.0001 Tc [(\(1\))-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 Tc 2.512 0 TD 357.557 625.823 m 14.3462 0 0 14.3462 320.94 401.988 Tm 1.1194 0 TD [(De�nition)-375.6(3.1.1)]TJ /F2 1 Tf 0.4587 0 TD -0.0001 Tc 14.3462 0 0 14.3462 410.265 660.4141 Tm 17.2155 0 0 17.2155 72 513.291 Tm 11.9551 0 0 11.9551 72 736.329 Tm (\))Tj 0 g 0.5101 0 TD )]TJ [(con)26.1(v)-13(\()]TJ 0.2779 Tc 226.093 654.17 l [(c)50.2(o)0(mbinations)]TJ 112.707 654.17 l 0.5001 0 TD 0 Tc (S)Tj 0.3337 0 TD ()Tj 0 g (If)Tj 12.4077 0 TD (\). 0.0001 Tc /F7 1 Tf /F4 1 Tf /F4 1 Tf /F3 1 Tf /F3 1 Tf [(used)-436.5(to)-436.1(giv)26.1(e)-436.5(a)-436.5(f)0(airly)-436.5(short)-436.4(p)-0.1(ro)-26.2(of)-436.4(of)-436.4(a)-436.1(g)-0.1(eneralization)-436.5(o)-0.1(f)]TJ (i)Tj /F2 1 Tf ()Tj 0 0 1 rg 1.0559 0 TD 0 Tc >> endobj (i)Tj 11.9551 0 0 11.9551 161.928 572.1901 Tm 0.6669 0 TD [(de�nitions)-301.8(ab)-26.1(out)-301.8(cones. /F3 1 Tf 0 -1.2052 TD 0.0001 Tc )Tj /F4 1 Tf /F2 1 Tf /F9 20 0 R /F2 1 Tf /F4 1 Tf (m)Tj /F9 1 Tf 0.4164 0 TD /F5 1 Tf /F4 1 Tf 0 g /F4 1 Tf [(hul)-50.1(l)]TJ /F4 1 Tf 0 Tc 226.093 685.464 200.694 710.863 169.4 710.863 c 0 Tc /Length 3049 4.6415 0 TD 0.0001 Tc (m)Tj 11.9551 0 0 11.9551 306.315 613.9529 Tm -10.1165 -1.2057 TD 0.0001 Tc 20.6626 0 0 20.6626 351.477 268.7791 Tm [(are)-301.9(t)0(he)-301.9(\(closed\))-301.9(half)-301.8(s)0(paces)-301.9(asso)-26.2(ciated)-301.9(with)]TJ Examples and properties • solution set of linear equations Ax = b (affine set) -19.4754 -1.2057 TD ()Tj Probably, the flrst topic who make necessary the encounter with this theory is the graphical analysis. /F2 1 Tf 0.0001 Tc 226.093 597.477 l (a)Tj [(union)-375.5(of)-375.4(triangles)-375.5(\(including)-375.5(in)26(terior)-375.5(p)-26.2(oin)26(ts\))-375.5(whose)-375.5(v)26.1(er-)]TJ 13.9283 0 TD (,)Tj /F4 1 Tf 345.472 612.855 344.535 613.792 343.38 613.792 c /F8 1 Tf (\()Tj ()Tj 1.6025 0 TD Optimization Techniques and Applications Convex sets & convex … )]TJ 20.6626 0 0 20.6626 404.523 652.368 Tm 0 Tc endobj (E)Tj 2.6997 0 TD )]TJ /F4 1 Tf -13.8787 -1.2052 TD 7.8467 0 TD << /F2 1 Tf [(W)78.6(e)-290.6(get)-290.5(t)0(he)-290.1(feeling)-290.6(t)0(hat)-290.5(triangulations)-290.1(pla)26.1(y)-290.6(a)-290.1(crucial)-290.5(r)0(ole,)]TJ 0 Tc ()Tj 1.2087 0 TD )Tj /F2 1 Tf (a)Tj 0 Tc The set X is open if for every x ∈ X there is an open ball B(x,r) that entirely lies in the set X, i.e., for each x ∈ X there is r > 0 s.th. ()Tj /F4 1 Tf (S)Tj 3.9516 0 TD 1.0611 0 TD /F8 16 0 R 0 Tc /F2 1 Tf (I)Tj 0 Tc /F5 1 Tf 0 Tc 1.0855 0 TD /F5 1 Tf << 0.3541 0 TD 0.3541 0 TD /GS1 gs (2)Tj 0.1667 Tc /F2 1 Tf -22.3501 -1.2052 TD [(is)-202.5(con)26(v)26.1(ex,)-222.1(and)-202.2(the)-202.6(e)0(n)26(tire)-202.6(ane)-202.1(space)]TJ -22.0407 -1.2052 TD 357.557 597.477 m 14.3462 0 0 14.3462 134.721 433.2001 Tm /F4 1 Tf 1.0689 0 TD [(can)-377.2(b)-26.1(e)-377.6(w)-0.1(ritten)-377.2(as)-377.1(a)-377.2(c)0.1(on)26.1(v)26.2(e)0.1(x)-377.2(c)0.1(om-)]TJ /F5 1 Tf /F3 1 Tf 0.0001 Tc (i)Tj /F2 1 Tf ({)Tj 0.3541 0 TD 0.0001 Tc (E)Tj /F2 1 Tf 0.6669 0 TD 0.6991 0 TD 0.0001 Tc /F2 1 Tf )-467.2(When)]TJ -18.5408 -1.2057 TD (X)Tj [(Colorful)-349.8(Car)50.1(a)-0.1(th)24.8(�)]TJ 0 Tc 50 0 obj 0.0001 Tc 0.0782 Tc [(is)-251.8(the)-251.8(s)0(mallest)-251.3(ane)-251.8(set)-251.8(con)26(t)0(ain-)]TJ /F5 1 Tf /F5 1 Tf 20.6626 0 0 20.6626 72 518.709 Tm 0 Tc 1.143 0 TD /F2 1 Tf [(EODOR)81.5(Y)0(�S)-326.3(THEOREM)]TJ ET /F2 1 Tf 1.0559 0 TD ()Tj -22.3781 -1.7841 TD /F4 1 Tf /F2 1 Tf << >> ()Tj /F2 1 Tf /F4 1 Tf 0 Tc [(. [(\))-310(f)0.1(or)-310.5(all)]TJ 0 Tc 0 Tc 2.262 0 TD 0 Tc 5.5698 0 TD 6.5752 0 TD /F4 1 Tf (. /F4 1 Tf 0 0 1 rg stream /F5 1 Tf for all z with kz − xk < r, we have z ∈ X Def. /F2 1 Tf /F3 1 Tf ([)Tj 0.0588 Tc /F7 1 Tf 0.0001 Tc 20.6626 0 0 20.6626 195.444 292.4041 Tm 5.5685 0 TD (. /F7 1 Tf -14.6327 -1.2052 TD 1.494 w ()Tj 0.6669 0 TD (I)Tj [(eo)50.1(dory�s)-249.8(T)0.1(he)50.2(or)50.2(em,)-270.1(R)50.1(adon)100.1(�s)-249.8(The-)]TJ /F2 1 Tf 1.1425 0 TD 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- 0.3338 0 TD (? 1.9728 0 TD 14.3462 0 0 14.3462 190.152 289.299 Tm (+1)Tj -0.0001 Tc 0 Tc A convex set S is a collection of points (vectors x) having the following property: If P 1 and P 2 are any points in S, then the entire line segment P 1-P 2 is also in S.This is a necessary and sufficient condition for convexity of the set S. Figure 4-25 shows some examples of convex and nonconvex sets. [(consists)-322.3(of)]TJ (is)Tj -16.7185 -2.9069 TD (called)Tj -21.4158 -1.2052 TD 0.0001 Tc endobj (i)Tj /F4 7 0 R 2.1361 0 TD /F4 1 Tf 0 Tc (f)Tj /ProcSet [/PDF /Text ] 0 Tc (H)Tj /F4 1 Tf /F4 1 Tf /F8 16 0 R 0.0001 Tc (�)Tj 3.888 0 TD ��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��~$���/w��/b����4�za�!�AY�[�|�jm�J�Y,�� ��Fw������Q��J,T?~����w9�(�6S�l�Kӊ��@���MQ*WH�[��eA�����{���7�*�^u6��zGJ�)J�������e�Vjg[%� ��T�Q� l��!ǪE�6��rc`�m��>[�[[u�1���%#�@y{縎���MQ--�@v�iSPsHf�i�܌� F�BQ��oʮ�K�N�ߺ���Q��s�+#z��ʁ�f�\TЧ%��D�"�%�J���h4>��ַ�H�3��{]cY��[���o�>N�20ׁ��:��#�}�&|�@]�;�3��B_�)�kҋIwA����Z)�H�`¦�m�B�А_Ŭ�R�D�ާ������y�H2�*E;��$����S?ܭ��M��/�ݚ�ڤU�E)����Y�����դښ�(�pǪ����� �3��.`*��s2o0�)BkbL���i7���P-�/PI3e��\��Xޗ�����j�r�S�ٓ݊ (S)Tj 20.6626 0 0 20.6626 333.045 663.519 Tm 20.6626 0 0 20.6626 72 702.183 Tm [(\))-327.9(and)]TJ 1.8726 0 TD 0.6669 0 TD 5.1.4.1 Convex hull representation Let C Rnbe a closed convex set. [(. /F5 1 Tf /F4 1 Tf ()Tj (i)Tj /F2 1 Tf 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. -18.4184 -2.3625 TD 1.782 0 TD /F2 1 Tf (101)Tj /F1 4 0 R (+)Tj [(Con)26(v)26.1(ex)-355.5(sets)-355.4(pla)26.1(y)-355.5(a)-355.9(v)26.1(ery)-355.5(imp)-26.2(o)-0.1(rtan)26.1(t)-355.4(role)-355.5(in)-355.9(geometry)78.3(. /F5 8 0 R 0 J 0 j 1.494 w 10 M []0 d ()Tj 2.4118 0 TD 7.2429 0 TD 20.6626 0 0 20.6626 421.299 663.519 Tm 20.6626 0 0 20.6626 201.249 333.1561 Tm 0.6943 0 TD BT 14.3462 0 0 14.3462 521.019 206.5711 Tm 0.389 0 TD 2.2415 0 TD /F3 1 Tf 0.4504 Tc 20.6626 0 0 20.6626 417.555 258.078 Tm ()Tj /F2 1 Tf [(tices)-301.9(b)-26.2(elong)-301.9(to)]TJ /F2 1 Tf (+)Tj -19.2104 -3.6688 TD ({)Tj 0.0001 Tc )-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 /F7 1 Tf (If)Tj Formally, a set CˆRN is said to be convex if for any x 1 and x 2 in Cthe point x 1 + (1 )x 2 2Cfor any 2[0;1]. /F2 1 Tf -2.4898 -2.261 TD [(,)-349.8(and)]TJ /F2 1 Tf • A polyhedral convex set is characterized in terms of a finite set of extreme points and extreme directions • A real-valued convex function is continuous and has nice differentiability properties • Closed convex cones are self-dual with respect to polarity • Convex, lower semicontinuous functions are self-dual with respect to conjugacy /F9 1 Tf (S)Tj 11.9551 0 0 11.9551 72 736.329 Tm ()Tj /F5 8 0 R /F7 10 0 R 8.1141 0 TD 0.0001 Tc An important method of constructing a convex set from an arbitrary set of points is that of taking their convex hull (see Fig. (E,)Tj /F4 1 Tf /F2 1 Tf -13.1817 -1.2057 TD 3.5383 0 TD [(. 0 g 1.0628 0 TD 0.6669 0 TD /F2 1 Tf 0 Tc (c)Tj /F4 1 Tf 9.8368 0 TD (I)Tj 0.9417 0 TD 1.0789 0 TD 3.6454 0 TD 391.038 676.846 l -0.0261 Tc (a)Tj (a)Tj 4.7087 0 TD /F3 6 0 R (I)Tj [(p)50.1(oints)-350(of)]TJ 0.4587 0 TD ET ()Tj 20.6626 0 0 20.6626 182.34 663.519 Tm [(hul)-50.1(l)]TJ By applying this property several times, we observe that a convex set Cmust contain any convex … 13.4618 0 TD 0.9274 0 TD /F2 1 Tf )-435.6(F)74.9(or)-306.5(any)-306.8(p)50.1(oint,)]TJ 14.9132 0 TD (> Thus [1;0]T is a direction of this convex set.57 4.7 An Unbounded Polyhedral Set: This unbounded polyhedral set has many 46 0 obj Nonlinear Programming, 2nd Edition, by Dimitri P. Bertsekas, 1999, ISBN 1-886529-00-0, 791 pages 6. (and)Tj endobj 0.0001 Tc 0.0001 Tc 0.0001 Tc (\012)Tj /F4 1 Tf 0.9448 0 TD )]TJ (1)Tj /F1 1 Tf (f)Tj 6.6699 0.2529 TD 0.0001 Tc (. 0.7884 0 TD [(v)26.1(e)0(x)-305.4(s)0(ubset,)]TJ /F4 1 Tf /F3 1 Tf 0 Tc [(of)-359.4(dimen-)]TJ [(c)50.1(onvex)-420.3(hul)-50.1(l)-420.4(of)]TJ [(p)-26.2(o)-0.1(in)26(ts)-301.9(in)]TJ /F4 1 Tf [(amoun)26.1(ts)-301.3(to)-301.8(the)-301.8(c)26.2(hoice)-301.8(of)-301.7(one)-301.8(of)-301.7(the)-301.8(t)26.2(w)26.1(o)-301.8(half-spaces. 0.4587 0 TD /GS1 11 0 R ()Tj 0 Tc /F2 1 Tf (+1)Tj f -22.2956 -1.2052 TD (i)Tj )Tj endobj /F5 1 Tf /F7 1 Tf 0.5894 0 TD 20.6626 0 0 20.6626 149.112 626.313 Tm 9.1752 0 TD (�s. ()Tj /F2 1 Tf 342.225 613.792 341.288 612.855 341.288 611.7 c (C)Tj 11.9551 0 0 11.9551 72 736.329 Tm (H)Tj Dynamic Programming and Optimal Control, Two-Volume Set, ... 1020 pag es 4. 0 0 1 rg (i)Tj BT 0.2779 Tc /F3 1 Tf (S)Tj 0.2731 Tc -6.969 -1.2052 TD /F2 1 Tf 20.6626 0 0 20.6626 346.563 407.8741 Tm /F4 1 Tf /F4 1 Tf /Length 5240 0.0001 Tc 20.6626 0 0 20.6626 232.173 292.4041 Tm 0 -2.3625 TD /F5 1 Tf 442.597 597.477 l ET The point is that a convex curve forms the boundary of a convex set. [(3.2. 0.7814 0 TD /F2 1 Tf /F4 1 Tf /F3 1 Tf /F6 1 Tf 1.4827 0 TD -20.2879 -1.2057 TD [(it)-310(is)-310.5(enough)-309.7(to)-310.1(assume)-310.1(that)]TJ /F7 1 Tf (m)Tj (H)Tj -0.1302 -0.2529 TD /F2 1 Tf 0.389 0 TD -8.4369 -1.2052 TD 1.1312 0 TD /F2 1 Tf /F4 1 Tf (S)Tj This provides a bridge between a geometric approach and an analytical approach in dealing with convex functions. /F2 1 Tf (de�ning)Tj 0.0001 Tc 0.3038 Tc 0.0001 Tc [(\)L)300.5(e)250.3(t)]TJ /F2 1 Tf stream -15.875 -1.2052 TD 14.3462 0 0 14.3462 360.378 433.2001 Tm (a)Tj (\))Tj -15.5744 -1.2057 TD 0.3337 0 TD 20.6626 0 0 20.6626 363.654 407.8741 Tm BT 14.3462 0 0 14.3462 141.597 623.217 Tm (f)Tj convex optimization, i.e., to develop the skills and background needed to recognize, formulate, and solve convex optimization problems. 0.0527 -0.7187 TD (f)Tj /F7 1 Tf /F4 1 Tf /F1 4 0 R 0 Tc [(has)-330.5(�nite)-330.5(supp)-26.1(ort)-330.1(\()0.1(all)]TJ 14.3462 0 0 14.3462 160.092 465.7891 Tm 0.2777 Tc /F3 1 Tf 4.8219 0 TD -4.4777 -2.2615 TD (I)Tj (i)Tj (i)Tj [(family)-342.4(of)-342.8(half-spaces)-342.4(asso)-26.2(ciated)-342.4(with)-342.4(h)26(y)-0.1(p)-26.2(erplanes)-342.4(p)-0.1(la)26.1(y)-342.4(a)]TJ /F4 1 Tf 0.8564 0 TD (a)Tj (\). /F4 1 Tf (i)Tj [(nonempty)-507.7(sub-)]TJ /F4 1 Tf /F5 1 Tf (S)Tj /F5 1 Tf )-558.9(T)0.1(he)-386.6(family)]TJ (E,)Tj a ne image of the convex set Snunder the a ne mapping A: x7!B 1 P k i=1 x iA i, i.e. -17.1657 -2.941 TD 0.0001 Tc 2.5634 0 TD /F3 1 Tf -21.7439 -2.5664 TD ()Tj (V)Tj 0 Tc 1.2153 0 TD In Euclidean space, a region is a convex set if the following is true. 0.3062 Tc (\))Tj /F5 1 Tf [(tion)-349.8(of)-349.8(the)]TJ 0 -1.2057 TD 0.876 0 TD 1.63 0 TD /GS1 gs ET (i)Tj 0.3541 0 TD /F8 1 Tf 0.5101 0 TD /F4 1 Tf << [(Given)-429.6(an)-429.2(ane)-429.4(sp)50(ac)50.1(e)]TJ (i)Tj /F2 1 Tf /F4 1 Tf /F4 1 Tf ET /F5 1 Tf 0.3541 0 TD 20.6626 0 0 20.6626 453.762 626.313 Tm [(if)-280.9(for)-280.5(a)-0.1(n)26(y)-280.6(t)26.1(w)26(o)-280.6(p)-26.2(oin)26(t)0(s)]TJ 0.2496 0 TD /F9 1 Tf 0.9361 0 TD /F2 1 Tf /F3 1 Tf /F4 1 Tf /F4 1 Tf (+1)Tj /F5 1 Tf (:)Tj /F2 1 Tf /F4 1 Tf (I)Tj /F5 1 Tf /GS1 gs (H)Tj 6.4502 0 TD 0.0001 Tc (with)Tj 13.4618 0 TD 1.1769 0 TD 1.1604 0 TD (\()Tj (f)Tj (of)Tj 0.3541 0 TD 112.707 654.17 l )-567.1(I)0(n)]TJ /F2 1 Tf 20.6626 0 0 20.6626 136.521 468.894 Tm 6.5822 0 TD /F2 1 Tf 5.9251 0.7501 TD . 0.2989 Tc De nition 2. In particular, one should be acquainted with the geometric connection between convex functions and epigraphs. 0.5893 0 TD 387.657 628.847 l (1)Tj /F5 1 Tf (=0)Tj /F5 8 0 R /F4 1 Tf -0.0001 Tc [(+)-268(2)0(,)-381.8(and)]TJ ()Tj 345.472 611.7 m [(a,)-166.6(b)]TJ (\)=)Tj (b)Tj /F11 1 Tf 0.389 0 TD [(=\()277.7(1)]TJ /F9 20 0 R 13.4618 0 TD (i)Tj /F4 1 Tf (V)Tj (�)Tj (S)Tj /F4 1 Tf (=)Tj endstream /F5 1 Tf /F4 1 Tf << (dimension)Tj /F5 1 Tf BT /F1 1 Tf (i)Tj 1.2715 0 TD )Tj ()Tj 0.2777 Tc 20.6626 0 0 20.6626 278.838 258.078 Tm 4.4007 0 TD /F4 1 Tf /F7 1 Tf 0.5893 0 TD 0.6608 0 TD 3 0 obj )Tj -14.9132 -1.2052 TD /F2 1 Tf /F2 1 Tf /F4 1 Tf )]TJ 5.0201 0 TD << (+)Tj [(=K)277.5(e)277.7(r)]TJ 226.093 597.477 l ()Tj /F4 1 Tf 0 -2.3625 TD (I,)Tj -20.6834 -1.2057 TD -0.0002 Tc (+)Tj 2.0838 0 TD 357.557 597.477 l 30 0 obj 1.1255 0 TD 20.6626 0 0 20.6626 293.463 243.8761 Tm [(eo)50.1(dory)-350.3(t)0.2(he)50.2(or)50.2(em)]TJ 20.6626 0 0 20.6626 157.986 333.1561 Tm 20.6626 0 0 20.6626 378.234 242.5891 Tm 14.3462 0 0 14.3462 78.633 411.6901 Tm (for)Tj [(\(wher)50.1(e)]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 /F3 1 Tf /F1 1 Tf /F3 1 Tf /Font << /F9 1 Tf /GS1 11 0 R /F7 1 Tf /F4 1 Tf 1 i [(a,)-166.6(b)]TJ /F3 1 Tf (S)Tj (i)Tj /F4 1 Tf /F4 1 Tf /F2 1 Tf 220.959 620.154 m /F5 8 0 R 0.2777 Tc /F5 8 0 R 1.6295 0 TD 0 0 1 rg 0 Tc /F5 1 Tf 0.0001 Tc 0 -1.2052 TD -18.9164 -1.2057 TD (�)Tj /F2 1 Tf 14.3462 0 0 14.3462 161.964 548.499 Tm (f)Tj 0.5893 0 TD /F5 1 Tf [(is)-267.9(a)-268.4(�)0.1(nite)-267.9(\(of)-267.8(i)0(n�nite\))-268.3(set)-267.9(of)-267.8(p)-26.2(o)-0.1(in)26(ts)-268.3(in)-268(the)-267.9(a)-0.1(ne)-267.9(p)-0.1(lane)]TJ (S)Tj 0 g /F2 1 Tf 0 g /F2 1 Tf 0.3338 0 TD ()Tj ()Tj /F2 1 Tf /F7 1 Tf 0 g (b)Tj 0.3509 Tc << /F3 1 Tf 11.9551 0 0 11.9551 289.53 684.819 Tm
2020 convex set pdf