/GS1 gs [(the)-301.4(union)-301.9(of)-301.4(tetrahedra)-301.5(\(including)-301.9(in)26(terior)-301.4(p)-26.2(oin)26(ts\))-301.9(whose)]TJ 0.9592 0 TD 0 -1.2052 TD 442.597 597.477 l [(whic)26.1(h)-301.9(i)0(s)-301.9(o)-0.1(f)-301.8(c)0(ourse)-301.9(true! 14.3462 0 0 14.3462 125.127 490.701 Tm 414.25 625.823 l [(])-205.1(i)0(s)-205.2(o)-0.1(ften)-204.8(used)-204.8(to)-205.2(denote)-204.8(t)0(he)-204.8(line)-205.2(segmen)26.1(t)]TJ /F2 1 Tf /F2 1 Tf /F3 1 Tf 0 Tc 0.7836 0 TD 1 i 1.9728 0 TD 0.0001 Tc 14.3462 0 0 14.3462 244.179 538.1671 Tm (+)Tj [(,)-363.7(w)-0.2(here)]TJ /F2 1 Tf (\()Tj 0 0 1 rg /F2 1 Tf 0.8886 0 TD 0.0001 Tc (q)Tj BT /F7 1 Tf 0 Tc -2.4898 -2.261 TD /F4 1 Tf (\012)Tj << (b)Tj [(The)-247.9(e)0(mpt)26.1(y)-247.9(set)-248.3(is)-248.3(trivially)-248.3(con)26(v)26.1(ex,)-258.7(e)0(v)26.1(ery)-247.9(one-p)-26.2(oin)26(t)-247.8(set)]TJ 1.9745 0 TD (V)Tj 0.0001 Tc (b)Tj P Q Figure 1: A Convex Set P Q Figure 2: A Non-convex Set To be more precise, we introduce some de nitions. stream View Lecture3_ConvexSetsFuns.pdf from MAT MISC at National Taiwan University of Science and Technology. 14.3462 0 0 14.3462 187.893 330.0511 Tm 0.5893 0 TD /F7 1 Tf 14.3462 0 0 14.3462 340.056 265.683 Tm /F4 1 Tf 13.9283 0 TD /F2 1 Tf /F2 1 Tf /F3 1 Tf 3.4799 0 TD [(Basic)-374.7(P)-0.1(rop)-31.1(e)-0.1(rties)-375.4(of)-374.8(Con)31.3(v)31.3(e)-0.1(x)-375(S)0.1(ets)]TJ 20.6626 0 0 20.6626 483.327 677.28 Tm 0.849 0 TD /F5 1 Tf stream 20.6626 0 0 20.6626 295.929 258.078 Tm 0.0001 Tc 1.2549 0 TD 0 Tc (f)Tj (|)Tj 0.0001 Tc /F2 1 Tf 11.9551 0 0 11.9551 72 736.329 Tm -11.7021 -2.363 TD (m)Tj 14.3462 0 0 14.3462 290.637 254.973 Tm 0 0 1 rg 0.6608 0 TD 0.6943 0 TD 0.2778 Tc 0 Tc (i)Tj /F8 1 Tf 0 Tw /F4 1 Tf /F2 1 Tf 1.6025 0 TD 0.3338 0 TD /F3 1 Tf (mension)Tj /F4 1 Tf S 0.5893 0 TD (. (,)Tj 0.9974 0.7501 TD 0 Tc /F3 1 Tf 6.1156 0 TD 0.3338 0 TD ({)Tj /F2 1 Tf [(is)-202.5(con)26(v)26.1(ex,)-222.1(and)-202.2(the)-202.6(e)0(n)26(tire)-202.6(ane)-202.1(space)]TJ 0.3541 0 TD /F4 1 Tf (\)=)Tj The theory of convex functions is part of the general subject of convexity since a convex function is one whose epigraph is a convex set. 0.0001 Tc 2.8875 0 TD /F4 1 Tf 0.6608 0 TD 1.0689 0 TD (i)Tj /F2 1 Tf ()Tj /F4 1 Tf 0 Tc 0.6699 0 TD 14.3462 0 0 14.3462 216.234 261.6151 Tm 0.0001 Tc (\()Tj 5.1.4.1 Convex hull representation Let C Rnbe a closed convex set. /GS1 11 0 R -0.0003 Tc 1.525 0 TD /F4 1 Tf /F2 1 Tf 14.7128 0 TD /F7 1 Tf 14.3462 0 0 14.3462 356.058 239.493 Tm (i)Tj /F2 1 Tf [(spanned)-266.1(b)26.1(y)]TJ 0.4587 0 TD ()Tj ()Tj 0 -1.2052 TD 0 Tc /F4 1 Tf -0.0001 Tc -0.0001 Tc /F4 1 Tf 0.446 Tc /F7 1 Tf 0 g (i)Tj [(to)-452.2(the)-452.1(s)0.1(et)-452.1(of)-452.2(p)50(o)-0.1(sitive)-452.1(c)50.1(o)-0.1(mbinations)-451.6(o)-0.1(f)-451.8(families)-451.6(o)-0.1(f)]TJ (})Tj /F2 1 Tf /F4 7 0 R 0 Tc 12.9565 0 TD /F4 1 Tf /F4 1 Tf 20.6626 0 0 20.6626 258.93 195.0601 Tm [(p)-26.2(o)-0.1(in)26(ts)-301.9(in)26(v)26.1(o)-0.1(lv)26.1(ed)-301.9(in)-301.9(the)-301.9(c)0(on)26(v)26.1(e)0(x)-301.5(c)0(om)25.9(binations? (and)Tj 0.3338 0 TD /F2 1 Tf -21.4158 -1.2052 TD (f)Tj (that)Tj /F4 1 Tf [(,)-558(o)0(f)-516.6(di-)]TJ (i)Tj 0 Tc 6.5504 0 TD 0.0001 Tc 0.9282 0 TD 0.3541 0 TD /F4 1 Tf (i)Tj (101)Tj 0 g /F7 10 0 R /F5 1 Tf [(is)-420.8(con)26(v)26.1(ex)-420.4(if)-420.3([)]TJ /F2 1 Tf (with)Tj The support line at angle 19 for the closed and bounded 2D set S is given by where L&l) = {x E lR2 1 x*ttl = h(8)}, (1) h(8) = sup XTW. 0.0527 -0.7187 TD /F3 1 Tf 20.6626 0 0 20.6626 413.829 701.0491 Tm [(is)-202.5(of)-202.5(course)-202.1(con)26(v)26.1(ex. )-581(Instead,)-375(w)26(e)-359.8(refer)-360.2(the)-360.2(reader)-359.8(t)0(o)-360.3(M)-0.2(atousek)]TJ 0.0001 Tc 0 Tc )-558.9(T)0.1(he)-386.6(family)]TJ 1 0 0 RG /F2 1 Tf 0 -1.2052 TD 0 Tc 18 0 obj /F4 1 Tf (. /F1 1 Tf 0 -2.3625 TD 0.2836 Tc /F2 1 Tf (=0)Tj /F3 1 Tf (,H)Tj /F5 1 Tf 0.7836 0 TD 0.6669 0 TD (f)Tj [(,)-331.4(and)-325.9(the)-325.4(c)26.1(hoice)-325.8(of)-325.3(a)-325.9(s)0(p)-26.2(eci�c)]TJ (i)Tj -22.3781 -1.7841 TD 6.6218 0 TD )-406.2(B)0.2(y)-302.3(lemma)-301.4(3.1.2,)]TJ (\()Tj >> 6.5822 0 TD 0.0001 Tc >> endobj (�)Tj /F5 8 0 R 0.389 0 TD /F5 1 Tf /F2 1 Tf /F2 1 Tf /F4 1 Tf 19 0 obj 3.8 0 TD 0 Tc 20.6626 0 0 20.6626 120.879 590.4661 Tm 0 -1.2057 TD ()Tj 0.8564 0 TD (})Tj 425.933 611.65 l /F2 1 Tf 220.959 620.154 m 20.6626 0 0 20.6626 221.58 541.272 Tm 4.0253 0 TD /F4 1 Tf 0 -1.2052 TD -0.0001 Tc ({)Tj 0 Tc 0 Tc 0.5893 0 TD -0.0001 Tc 2.5835 0 TD (i)Tj [(c)50.1(onvex)-420.3(hul)-50.1(l)-420.4(of)]TJ /F4 1 Tf CONVEX SET RECONSTRUCTION 415 quantities precisely. 0.7392 0 TD 1.0358 0 TD /F4 1 Tf [(is)-353.6(an)26.1(y)-353.7(nonconstan)26.1(t)-353.6(ane)]TJ 0 Tc 0.3337 0 TD 0.5798 0 TD Convex combination and convex hull convex combination of x1,. /F4 1 Tf /F2 1 Tf -5.1486 -2.8447 TD 13.4618 0 TD 11.9551 0 0 11.9551 161.928 572.1901 Tm /F4 1 Tf /F3 1 Tf 0.8716 0 TD 0.5314 0 TD (i)Tj 0 Tc 0.5549 0 TD 14.9132 0 TD 0 Tc ()Tj 0 Tc [(c)50.2(one)]TJ /F3 6 0 R /F8 1 Tf 1.0789 0 TD [(\). /F5 1 Tf /F4 1 Tf ()Tj /F4 1 Tf -0.0003 Tc << /F5 1 Tf /F2 1 Tf 9.6003 0 TD 0.0001 Tc (C)Tj /F5 1 Tf (\()Tj 1.0855 0 TD (�)Tj [(eo)50.1(dory)-350.3(t)0.2(he)50.2(or)50.2(em)]TJ 0.5798 0 TD /F2 1 Tf [(\(wher)50.1(e)]TJ /F4 1 Tf (+)Tj 0.2777 Tc 442.597 597.477 m 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. /F4 1 Tf stream -15.875 -1.2052 TD /F7 1 Tf (f)Tj /F2 1 Tf (\()Tj (v)Tj [(smallest)-446.5(con)26(v)26.1(ex)-446.1(set)-446(con)26(t)0(aining)]TJ 0 g 0 Tc [(p)50.1(oints)-350(of)]TJ 0 Tc 430.492 611.7 m 0 g 0.6608 0 TD /Length 5240 (. /F3 6 0 R (E)Tj 0.7919 0 TD ()Tj /F4 1 Tf /F4 1 Tf (b)Tj /GS1 11 0 R /F3 1 Tf In general cone is not supposed to -19.4754 -1.2057 TD /F2 1 Tf 11.3505 0 TD /F3 1 Tf /F4 1 Tf 6.6161 0 TD 20.6626 0 0 20.6626 379.566 407.8741 Tm (\()Tj 0.3038 Tc << 3.9573 0 TD 1.2216 0.7187 TD 1.0559 0 TD /F2 5 0 R 0.2775 Tc -8.4369 -1.2052 TD /F9 1 Tf 0.0001 Tc 0.612 0 TD 20.6626 0 0 20.6626 453.51 375.2401 Tm 0 -1.2057 TD 0.7836 0 TD 14.3462 0 0 14.3462 187.416 587.3701 Tm 226.093 654.17 l [(Although)-266.1(Theorem)-266.3(3.2.6)-266.5(i)0.1(s)-266(not)-266.5(hard)-266.1(t)0.1(o)-266.5(p)0(ro)26.2(v)26.2(e)0.1(,)-273.4(w)26.1(e)-266.1(w)-0.1(ill)-266.4(not)]TJ /F4 1 Tf 9.0336 0 TD 20.6626 0 0 20.6626 72 702.183 Tm [(is)-250.2(any)-250.1(c)50.2(o)0(mp)50.1(act)-250.3(subset)-251.2(o)0(f)]TJ 0.0001 Tc endobj 14.3462 0 0 14.3462 484.578 240.78 Tm (L)Tj /F2 1 Tf /F2 1 Tf (�)Tj /Font << /F5 1 Tf 0.9622 0 TD >> 31.1377 0 TD 3.2007 0 TD [(\(namely)78.4(,)-393.2(t)0.1(he)]TJ 0.9417 0 TD (=1)Tj /F2 1 Tf /F2 1 Tf 0.9539 0 TD 31.1377 0 TD )-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 [(p)50.1(o)0(lyhe)50.2(dr)50.2(al)-350.1(c)50.2(one)]TJ -21.5619 -1.2052 TD 2.4118 0 TD (i)Tj /F1 1 Tf (V)Tj /F5 1 Tf 0.3338 0 TD /F4 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 0.6608 0 TD 0.3499 Tc 0.0001 Tc (i)Tj 0.0001 Tc (S)Tj (H)Tj (b)Tj >> (\()Tj 0.0001 Tc 0.0001 Tc 0.0001 Tc (0)Tj (I)Tj /F9 1 Tf 0.2781 Tc (|)Tj (m)Tj (E)Tj [(,)-448.7(for)]TJ 0 Tc /F4 1 Tf 0 G (+)Tj 14.3462 0 0 14.3462 397.629 341.274 Tm /F2 1 Tf /F4 1 Tf /F4 1 Tf 0.3391 Tc -0.0001 Tc 430.492 612.855 429.555 613.792 428.4 613.792 c (�)Tj 0 Tc (\()Tj /F4 1 Tf /F5 1 Tf -21.7941 -1.2057 TD 0 g Lecture 2: Basic Convex Analysis { August 25 2-5 2.1.4 Geometry of Convex Sets Convex sets are special because of their nice geometric properties. 391.038 591.807 l 0.4504 Tc 0 Tc 20.6626 0 0 20.6626 72 518.709 Tm 0 Tc BT (a)Tj (1)Tj /F8 16 0 R -22.0456 -2.3625 TD )-435.6(F)74.9(or)-306.5(any)-306.8(p)50.1(oint,)]TJ /F2 1 Tf (])Tj 0.0229 Tc 2.2019 0 TD 20.6626 0 0 20.6626 404.523 652.368 Tm /F4 1 Tf [(. 112.707 625.823 m 20.6626 0 0 20.6626 170.811 468.894 Tm 0 Tc << 0.6991 0 TD 10.0402 0 TD 0 g /F2 1 Tf [(There)-212.2(is)-212.6(also)-212.2(a)-212.6(stronger)-212.1(v)26.1(ersion)-212.6(o)-0.1(f)-212.1(T)-0.2(heorem)-212.3(3.2.6,)-230.4(in)-212.2(whic)26.1(h)]TJ 0 g /F4 1 Tf 0.3062 Tc /F4 1 Tf -19.6267 -1.2052 TD /GS1 gs (H)Tj 15.2007 0 TD /F2 1 Tf >> ()Tj /F1 1 Tf 0.0001 Tc (de�ning)Tj /F4 1 Tf [(�nite)-366.3(set)-365.9(of)-366.3(cardinalit)26.1(y)]TJ [(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 /F2 1 Tf (R)Tj /F5 1 Tf -0.0003 Tc (a)Tj TODO). 1.8064 0 TD /F3 1 Tf 6.6699 0.2529 TD /F2 1 Tf 0.7836 0 TD 0 Tc /F7 1 Tf /F2 1 Tf 0 G (E)Tj /F7 10 0 R /F4 1 Tf -12.5597 -1.2052 TD )]TJ /F4 1 Tf (\()Tj 0.9975 0 TD (\(with)Tj (v)Tj /F3 1 Tf (,)Tj (\()Tj /F4 1 Tf )]TJ 0.5763 0 TD 20.6626 0 0 20.6626 249.741 576.498 Tm (a)Tj 11.9551 0 0 11.9551 72 736.329 Tm -0.1302 -0.2529 TD ()Tj 6 0 obj 0 Tc 2.262 0 TD [(,)-315.4(t)0.2(he)-306.5(c)50.2(one,)]TJ [(ve)50.1(ctors)-306.9(i)-0.1(n)]TJ (H.)Tj 0.9443 0 TD 0.9705 0 TD /F4 1 Tf (+1)Tj /F4 1 Tf 0 g /F4 1 Tf if x2C, then tx2Cfor any t>0. (1)Tj /F2 1 Tf (E)Tj /F3 1 Tf 9.1752 0 TD )-762.6(CARA)81.1(TH)]TJ /F4 1 Tf 0.9443 0 TD /F3 1 Tf 0.7836 0 TD 0 g /F4 1 Tf /GS1 11 0 R (and)Tj 0.9861 0 TD (E)Tj >> /F9 1 Tf ()Tj 0.585 0 TD 6.675 0 TD Nonetheless it is a theory important per se, which touches almost all branches of mathematics. /ProcSet [/PDF /Text ] (I)Tj [(S,)-384.2()]TJ /F2 1 Tf /F3 1 Tf /F6 9 0 R 0.5893 0 TD /F1 4 0 R /F2 1 Tf (xa)Tj 1.1451 0 TD 0.0001 Tc 0 0 1 rg 0 g (a)Tj 2.4898 0 TD 0 Tc (|)Tj (a)Tj BT 387.657 636.416 l 0.6991 0 TD 40 0 obj /F2 1 Tf >> (96)Tj 0 Tc (=\()Tj -21.7619 -1.2057 TD /F4 1 Tf (S)Tj /F2 1 Tf 1.065 0 TD /F2 1 Tf /Font << 20.6626 0 0 20.6626 199.431 541.272 Tm 1.2153 0 TD [(\))-327.9(and)]TJ [(,o)273(r)]TJ /F2 1 Tf << /F2 1 Tf 0.9822 0 TD /F2 1 Tf 2.7455 0 TD 1.6896 0 TD 0.3776 Tc << (\()Tj 0.3338 0 TD /F5 1 Tf [(in�nite\))-301.9(of)-301.8(con)26(v)26.1(ex)-301.9(sets)-301.9(is)-301.9(con)26(v)26.1(ex. 14.3462 0 0 14.3462 190.152 289.299 Tm T* /F4 1 Tf ET /F4 1 Tf (q)Tj BT endobj 20.6626 0 0 20.6626 365.103 590.4661 Tm (with)Tj /GS1 gs 0.3499 Tc [(asserts)-461.7(that)-462.1(it)-462.1(is)-461.7(enough)-461.8(to)-462.2(consider)-461.7(con)26(v)26.1(ex)]TJ >> /F8 16 0 R (i)Tj 0.2496 0 TD 45 0 obj 14.3462 0 0 14.3462 170.163 330.0511 Tm /F4 1 Tf /F4 1 Tf 3.6454 0 TD BT 2.8204 0 TD stream (m)Tj [(,)-349.8(and)]TJ -20.6884 -1.2052 TD 0 Tc [(+)-286.4(2)-0.1(,)-414.2(t)0(he)-392(p)-26.2(o)-0.1(in)26(ts)]TJ 0 -2.7349 TD [(the)-324(c)50.1(onvex)-323.6(hul)-50.1(l)]TJ [(,)-375.8(recall)-361.5(that)-361.5(a)-361.1(s)0(ubset)]TJ 11.9551 0 0 11.9551 72 736.329 Tm 1.1534 0 TD 0.3541 0 TD -7.9956 -2.363 TD 0.0041 Tc [(only)-376.7(d)0(ep)-26.1(ends)-376.2(on)-376.8(the)]TJ [(\))-310(f)0.1(or)-310.5(all)]TJ (H)Tj 0 0 1 rg )Tj ET 20.6626 0 0 20.6626 157.986 333.1561 Tm /F4 1 Tf /ExtGState << @m1�%I�Ƙ[�ǝD /F2 1 Tf 0 Tc << 6.5752 0 TD )-813.2(In)-437.4(case)-437.3(1,)-471.2(assuming)]TJ /F4 1 Tf (\()Tj 220.959 591.807 l [(is)-301.9(con)26(v)26.1(ex. 0.3541 0 TD 1.0855 0 TD /F4 1 Tf (A)Tj 0.0001 Tc 0.0001 Tc 14.3462 0 0 14.3462 431.712 526.593 Tm 4.8503 0 TD 0.3541 0 TD /Font << /F3 6 0 R 0.9443 0 TD (L)Tj /F2 1 Tf (i)Tj /F4 1 Tf /F2 1 Tf 0.7884 0 TD 226.093 685.464 200.694 710.863 169.4 710.863 c 14.3462 0 0 14.3462 517.824 540.5161 Tm 0.4587 0 TD 0.0002 Tc [(=)-328.6(0)-330.2(except)-330.2(f)0(or)-330.6(�nitely)]TJ (�)Tj 0.0001 Tc 14.3462 0 0 14.3462 153.135 638.9041 Tm (I)Tj 0.0001 Tc (m)Tj 0 -1.2057 TD Preface The theory of convex sets is a vibrant and classical field of modern mathe-matics with rich applications in economics and optimization. (subsets)Tj << (\). /F3 1 Tf ()Tj 0.3541 0 TD (i)Tj 0.514 0 TD /F2 1 Tf -14.5816 -1.2052 TD /F2 1 Tf (cone\()Tj 0.72 0 TD /F4 1 Tf /F9 20 0 R /F9 20 0 R 0.1666 Tc 20.6626 0 0 20.6626 443.367 529.6981 Tm (E)Tj /F4 1 Tf 4.1503 0 TD /F2 5 0 R /F5 1 Tf 1.4566 0 TD 20.6626 0 0 20.6626 182.34 541.272 Tm [(v)26.1(e)0(x)-305.4(s)0(ubset,)]TJ 0.849 0 TD /F4 7 0 R 9.9253 0 TD (I)Tj [(,)-306.1(i)0.1(s)-305.7(t)0.1(he)-305.7(dimension)]TJ 6.6699 0.2529 TD 0 Tc ()Tj /GS1 11 0 R /F2 1 Tf )Tj 20.6626 0 0 20.6626 388.278 493.7971 Tm 2.7751 0 TD be identi ed with certain convex subsets of Rn+1, while convex sets in Rn can be identi ed with certain convex functions on Rn. [(is)-251.8(the)-251.8(s)0(mallest)-251.3(ane)-251.8(set)-251.8(con)26(t)0(ain-)]TJ (f)Tj 0.9448 0 TD 0 -1.2052 TD -6.969 -1.2052 TD -0.0002 Tc S 0 Tc 0.3541 0 TD (j)Tj 0 g 4.0627 0 TD That is, coX:= X … (S)Tj /F4 1 Tf (})Tj 20.6626 0 0 20.6626 201.249 333.1561 Tm /F2 1 Tf /GS1 gs 0.6608 0 TD 14.3462 0 0 14.3462 501.534 697.953 Tm (\()Tj 0.2779 Tc (100)Tj /F4 1 Tf 0.6669 0 TD 0 Tc [(=K)277.5(e)277.7(r)]TJ 357.557 597.477 m /F2 1 Tf /F2 1 Tf [(This)-339.3(theorem)-339.5(due)-339.8(to)-339.4(B�)]TJ /F2 1 Tf /F2 1 Tf (where)Tj 0 Tc /F4 1 Tf ()Tj [(if)-280.9(for)-280.5(a)-0.1(n)26(y)-280.6(t)26.1(w)26(o)-280.6(p)-26.2(oin)26(t)0(s)]TJ 8.4369 0 TD /F3 1 Tf -0.0001 Tc 0 Tc 14.3462 0 0 14.3462 233.586 433.2001 Tm 0.315 Tc /F6 1 Tf 0.6669 0 TD (q)Tj [(eo)50.1(dory�s)-249.8(T)0.1(he)50.2(or)50.2(em,)-270.1(R)50.1(adon)100.1(�s)-249.8(The-)]TJ (I,)Tj 0 g 1.386 0 TD [(of)-251.6(a)-251.7(s)0.1(ubset,)]TJ So, any shape which is concave, or has a hollow, cannot be a convex set. /F4 1 Tf 14.3462 0 0 14.3462 244.179 660.4141 Tm 5.5685 0 TD (=1)Tj [(Figure)-325.9(3.2:)-436.4(The)-325.9(t)27(w)27.4(o)-326.5(half-spaces)-326.7(determined)-325.5(b)26.8(y)-326.4(a)-326.5(h)26.8(y)0.4(p)-27.4(e)0.1(rplane,)]TJ [(or)50.2(em,)-349.8(and)-349.7(Hel)-50(l)0.1(y�s)-349.5(T)0.1(he)50.2(or)50.2(em)]TJ /F3 1 Tf For any two points inside the region, a straight line segment can be drawn. /F5 1 Tf 0.0002 Tc 0.0001 Tc /F4 1 Tf <> 0 Tc -9.8325 -1.2052 TD -9.6165 -2.3625 TD C= A (Sn) . 0.5893 0 TD (S)Tj /Font << 14.3462 0 0 14.3462 336.168 526.593 Tm (\))Tj 6.4502 0 TD /F6 1 Tf (+)Tj 0 Tw [(Giv)26.1(e)0(n)-323.7(a)-0.1(n)26(y)-323.3(v)26.1(ector)-323.6(s)0(pace,)]TJ -22.0415 -1.2057 TD [(con)26.1(v)-13(\()]TJ (\))Tj 1.386 0 TD )-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 /F8 16 0 R >> (. 5.2257 0 TD 0 Tc (S)Tj 0.0001 Tc 0.8886 0 TD /F5 1 Tf (f)Tj (. 0.3541 0 TD /F2 1 Tf (i)Tj /F9 1 Tf s 0 Tc /F5 1 Tf [(\)i)283.7(st)283.6(h)283.5(e)]TJ /F4 1 Tf /F4 1 Tf 14.3462 0 0 14.3462 109.458 587.3701 Tm 0.4999 0.95 TD (\))Tj 0.8563 0 TD Convex combination Definition A convex combinationof the points x1,⋅⋅⋅ ,xk is a point of the form 1x1 +⋅⋅⋅ + kxk, where 1 +⋅⋅⋅ + k = 1 and i ≥ 0 for all i = 1,⋅⋅⋅ ,k. A set is convex if and only if it contains every convex combinations 14.3462 0 0 14.3462 303.831 516.657 Tm 0 Tc 14.3462 0 0 14.3462 458.802 515.6041 Tm /F2 1 Tf 6.1448 0 TD (1)Tj /F2 1 Tf /F4 7 0 R 3.0212 0 TD << [(,s)315.1(p)365(a)314.9(n)314.8(n)314.9(e)365.1(d)8.3(b)315(y)]TJ /F4 1 Tf (and)Tj -0.0002 Tc /ProcSet [/PDF /Text ] x��TKs1��أ����8�. 1.4971 0 TD /F5 1 Tf 0 g 345.472 611.7 m /F2 1 Tf (S,)Tj f /F4 1 Tf )-681.6(S)-0.1(ince)]TJ (f)Tj /F2 1 Tf 14.3462 0 0 14.3462 225.432 548.499 Tm On the right, we are able to draw a number of lines between points on the graph which actually do dip below the graph. )-499.5(The)]TJ 17.5537 0 TD (0)Tj [(called)-301.9(a)]TJ 1607 (,...,a)Tj /F4 1 Tf -0.0001 Tc 357.557 625.823 l (+)Tj (of)Tj 0.4587 0 TD (q)Tj /F4 1 Tf 0.9361 0 TD S /F5 1 Tf /F4 1 Tf ()Tj 0.0001 Tc /F2 1 Tf 0 Tc [(p)50(o)-0.1(sitive)]TJ /F5 1 Tf (\()Tj [(,i)536.6(f)]TJ /F5 1 Tf /F2 5 0 R /F2 1 Tf /F5 1 Tf (m)Tj /F5 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 (i)Tj 0.8881 0 TD 14.3462 0 0 14.3462 131.229 465.7891 Tm [(c)50.2(onvex)-390.6(c)50.2(one)]TJ [(,)-349.8(s)0.2(o)-350.2(t)0.1(hat)]TJ /F4 1 Tf /F2 1 Tf /F6 1 Tf [(ened)-301.9(in)-301.9(t)26.1(w)26(o)-301.9(d)-0.1(irections:)]TJ 2.512 0 TD (1)Tj /F2 1 Tf (for)Tj 0.7836 0 TD /F2 1 Tf /F5 1 Tf 11.9551 0 0 11.9551 300.15 74.6401 Tm [(CHAPTER)-327.3(3. 8.6743 0 TD /ExtGState << /GS1 gs (i)Tj (\()Tj 7.3645 0 TD (H)Tj /F5 1 Tf 2.1366 0 TD ()Tj 0.0001 Tc (\()Tj 5.2234 -1.7841 TD [(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 /F5 8 0 R 0 Tc (i)Tj 0.8563 0 TD /F4 1 Tf 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]. /F4 1 Tf 0.6608 0 TD (If)Tj /F4 1 Tf >> (S)Tj [(ve)50.1(ctors)-350.5(i)-0.1(n)]TJ 0.9443 0 TD 14.3462 0 0 14.3462 478.044 674.175 Tm ()Tj /F2 1 Tf If every point on that segment is inside the region, then the region is convex. )]TJ /F4 1 Tf (H)Tj [(Theorem)-375.9(3.2.2)]TJ ()Tj 20.6626 0 0 20.6626 421.299 541.272 Tm [(3.2. )]TJ 1.8064 0 TD ()Tj 0.0001 Tc [(is)-323.5(e)50.1(q)0(ual)-324.1(t)0.1(o)-324.1(t)0(he)-323.6(set)-324.4(o)-0.1(f)-323.7(c)50.1(onvex)]TJ )Tj /F2 1 Tf 1.1116 0 TD 11.9551 0 0 11.9551 378.099 572.1901 Tm (E)Tj 1.6291 0 TD The easiest way to figure out if a graph is convex or not is by attempting to draw lines connecting random intervals. ()Tj /F3 1 Tf /F4 1 Tf ()Tj 20.6626 0 0 20.6626 336.042 576.498 Tm 1.1386 0 TD /F5 1 Tf /F3 1 Tf /F4 1 Tf 0 Tc 10.2528 0 TD /F2 1 Tf /F4 1 Tf [(Colorful)-349.8(Car)50.1(a)-0.1(th)24.8(�)]TJ 0.7919 0 TD /F1 1 Tf /F2 1 Tf /F2 5 0 R /F2 1 Tf ()Tj /F5 1 Tf 1.0559 0 TD 2.644 0 TD (. ()Tj • 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 0.2223 Tc 0 Tc 0 Tc 0.7366 0 TD 9.6504 0 TD /F2 1 Tf [(. ET 0.2779 Tc This note studies the definition and properties of convex sets, convex functions, and convex opti … 1.0559 0 TD 15.1802 0 TD /F3 1 Tf /F7 1 Tf [(Given)-516.7(any)-516.4(ve)50.1(ctor)-517(sp)50(ac)50.1(e,)]TJ 1 i /F4 1 Tf /F2 1 Tf [(Helly�s)-372(theorem)-371.8(kno)26.1(wn)-371.6(as)-372(Tv)26.1(erb)-26.2(erg�s)-372(theorem)-371.8(\(see)-372(Sec-)]TJ 20.6626 0 0 20.6626 107.847 436.3051 Tm 0 J 0 j 0.996 w 10 M []0 d /F3 6 0 R 0 -2.3625 TD /F4 1 Tf (j)Tj /F2 1 Tf 0 Tc (m)Tj 0 0 1 rg [(b)50.2(e)-306.9(any)-306.3(ane)-306.5(sp)50.1(ac)50.2(e)-306.9(o)0(f)-306.7(dimension)]TJ (v)Tj [(Theorem)-375.9(3.2.5)]TJ /F4 1 Tf 0 Tc /GS1 11 0 R [(is)-350.1(also)-349.8(c)50.1(o)-0.1(mp)50(act. [(b)50.2(e)-386.6(a)-386.3(family)-386(o)0(f)-386.4(p)50.1(oints)-386.6(i)0(n)]TJ /F4 1 Tf (. 0 Tc 7.9634 0 TD [(p)50(o)-0.1(sitive)]TJ /ExtGState << 0 g /F2 1 Tf /F2 1 Tf (\()Tj 3.3131 0 TD -21.7937 -1.2057 TD /F4 1 Tf 2.0207 0 TD endobj /F3 1 Tf (E)Tj 0.5893 0 TD 0.0001 Tc 5.5698 0 TD [(=\()277.7(1)]TJ (i)Tj [(Then,)-427.1(g)0(iv)26.2(en)-402(an)26.1(y)-402(\()0.1(nonempt)26.2(y)0(\))-401.9(s)0.1(ubset)]TJ /Length 4531 [(nonempty)-507.7(sub-)]TJ ()Tj endobj 0.6669 0 TD -14.6327 -1.2052 TD 0.2989 Tc 0 g (I)Tj 1.4566 0 TD /GS1 gs /F2 1 Tf (=0)Tj 0.6608 0 TD /F4 1 Tf 0.0002 Tc 0.3541 0 TD 0 -1.2057 TD /F2 1 Tf (\))Tj 1.6469 0 TD /F2 1 Tf 0.8564 0 TD (a)Tj endstream /F4 1 Tf x��X[��F}����aݑ�זP%%�צP�P�[`H/��+�fl���iY‚�H#it���0�F#?���e0������ �/c��v?� ��J#�9���A�`f��`�i�-���Wfv�`�ӳ�G3�hb��1'�M�����70)�`������qz��h��`��5��Rt���9�����I \�'p���C��ߩgHQB�+�OT�d ���ي���sn������ q#����O\��ǒ�iz�g͉���3[��B��6��L��r�`&n��~d����W�d�����G�j1A j0L> w�@��ѿ7aw��5�d��6��۵!�it�ӜܸT��fO�������< � o����p|���n��l�h�n���CA8���;�^�f��%k%�r�S�ٳ�\(�")�eHQ��9���BE��Q\�X2ʢz���r�4�����߳�Pz�7-�j. 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