*Wednesday, December 9th, 2020*

0.9448 0 TD -14.6327 -1.2052 TD (97)Tj 0 Tc stream /F4 1 Tf (f)Tj De nition 2. /Length 5240 /F4 1 Tf 0 Tc -19.5186 -1.2052 TD 0.7718 0 TD 1.0554 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 (\()Tj -0.1302 -0.2529 TD 0.3999 0 TD (S)Tj An optimal (global) solution of (1.1) is x∗ ∈ argmin x∈X f(x), (1.2) and this has the smallest value of the function f among all vectors in the set X. /F4 1 Tf /F1 1 Tf [(3.1)-1125(C)0.2(on)31.2(v)31.3(e)0(x)-374.4(S)-0.2(ets)]TJ /F2 1 Tf [(CHAPTER)-327.3(3. 0.8912 0 TD 5.8064 0 TD 0.5314 0 TD 11.3505 0 TD /F2 1 Tf (f)Tj /F9 1 Tf (i)Tj T* /F2 5 0 R [(of)-301.8(strictly)-301.9(less)-301.9(than)]TJ 1. (S)Tj /F4 1 Tf /F5 1 Tf /ExtGState << 429.555 609.608 430.492 610.545 430.492 611.7 c 112.707 597.477 m (0)Tj 20.6626 0 0 20.6626 72 702.183 Tm /F2 1 Tf (1)Tj The point is that a convex curve forms the boundary of a convex set. /F3 1 Tf (\()Tj /F2 1 Tf 14.3462 0 0 14.3462 389.178 649.272 Tm ()Tj 0.8886 0 TD (�)Tj (\)=)Tj /F4 1 Tf 0.3541 0 TD [(ev)26.1(ery)-298.9(p)-26.2(o)-0.1(in)26(t)]TJ 0.5798 0 TD [(consists)-322.3(of)]TJ (m)Tj /F7 1 Tf ()Tj 1.0554 0 TD [(a,)-166.6(b)]TJ /F2 1 Tf (S)Tj [(,)-493.6(let)]TJ 6.6699 0.2529 TD /F2 1 Tf 0.0002 Tc /F7 10 0 R /F3 1 Tf 0 g (b)Tj (i)Tj 0.889 0 TD (i)Tj (b)Tj [(is)-306.4(e)50.2(q)0.1(ual)]TJ Examples and properties • solution set of linear equations Ax = b (aﬃne set) /F4 1 Tf 15.2007 0 TD 1.596 0 TD 0.4586 1.7101 TD Nonetheless it is a theory important per se, which touches almost all branches of mathematics. @m1�%I�Ƙ[�ǝD 17.5537 0 TD 0.6669 0 TD 0 Tc 0.8768 0 TD 0.5001 0 TD 0.8563 0 TD 0.3541 0 TD 0 Tw 14.3462 0 0 14.3462 339.822 487.911 Tm /F4 1 Tf (�nite)Tj (,)Tj 14.3462 0 0 14.3462 194.139 538.1671 Tm 8.6743 0 TD endstream (\)=)Tj (\(b\))Tj 0.9274 0 TD [(EODOR)81.5(Y)0(�S)-326.3(THEOREM)]TJ %�쏢 S /F2 1 Tf 0.4587 0 TD 0.0001 Tc 0.5893 0 TD /F3 1 Tf /F4 1 Tf [(In)-244.9(case)-244.4(2,)-256.1(the)-244.8(t)0(heorem)-244.1(of)]TJ [(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 6.3273 0 TD /F4 1 Tf 11.9551 0 0 11.9551 306.315 613.9529 Tm (V)Tj << ()Tj /F4 1 Tf [(+)-268(2)0(,)-381.8(and)]TJ /F4 1 Tf (. 11.9551 0 0 11.9551 72 736.329 Tm [(of)-301.8(the)-301.9(s)0(mallest)-301.9(ane)-301.9(subset)]TJ /F4 1 Tf 1.0611 0 TD /F5 1 Tf (. /GS1 gs 0.0001 Tc 391.038 591.807 l 0.1666 Tc Convex combination and convex hull convex hull convS: set of all convex combinations of points in S convex combination of x 1,. . 0 0 1 rg /F5 1 Tf 20.6626 0 0 20.6626 169.488 551.595 Tm (a)Tj -13.7396 -1.2052 TD 0.9443 0 TD /GS1 gs /F4 1 Tf /F2 1 Tf 0 Tc 14.3462 0 0 14.3462 410.265 660.4141 Tm [(\). 4.4443 0 TD /F5 1 Tf 14.3462 0 0 14.3462 249.633 233.463 Tm 0 Tc [(\)\()446(o)445.9(r)]TJ /F4 1 Tf /F5 1 Tf [(,o)536.6(f)]TJ 0.7836 0 TD /F4 1 Tf /F2 1 Tf (f)Tj 14.3462 0 0 14.3462 153.135 638.9041 Tm >> (|)Tj 0 -1.2052 TD /F9 1 Tf 0 Tw ()Tj 0.0001 Tc 0 g /F8 16 0 R /F2 1 Tf endobj /F5 1 Tf convex set: contains line segment between any two points in the set x1,x2 ∈ C, 0≤ θ ≤ 1 =⇒ θx1+(1−θ)x2 ∈ C examples (one convex, two nonconvex sets) Convex sets 2–3. ([)Tj 7.9701 0 0 7.9701 299.232 683.028 Tm /F5 1 Tf /F4 1 Tf [(\),)-285.2(w)26(e)-280.5(can)-280.6(d)-0.1(e�ne)-280.5(the)-280.5(t)26.1(w)26(o)]TJ (subsets)Tj 0.6669 0 TD 20.6626 0 0 20.6626 378.234 242.5891 Tm /F2 1 Tf 14.3462 0 0 14.3462 343.818 380.181 Tm 1.2113 0.95 TD /F2 1 Tf [(3.1. (+1)Tj [(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 /F4 1 Tf (and)Tj [(m)25.9(u)-0.1(st)-391.6(b)-26.2(e)-392(anely)]TJ 8.0035 0 TD /F3 1 Tf 20.6626 0 0 20.6626 492.642 375.2401 Tm 0 Tc /F4 1 Tf 0.2783 Tc )-558.9(T)0.1(he)-386.6(family)]TJ 0 Tc /ProcSet [/PDF /Text ] /F4 1 Tf /F4 1 Tf [(,)-287.3(t)0.1(heorem)-283.7(3.2.2)-283.5(c)0.1(on�rms)-283.9(our)-283.5(in)26.1(tuition)-283.5(t)0.1(hat)]TJ /F2 1 Tf /F5 1 Tf (96)Tj (S)Tj /F4 1 Tf /F1 1 Tf (\))Tj /F4 1 Tf 20.6626 0 0 20.6626 72 702.183 Tm /F4 1 Tf 0 Tc /F4 1 Tf [(,)-315.4(t)0.2(hat)-306.9(is,)]TJ (S)Tj [(Theorem)-375.9(3.2.6)]TJ 0.1667 Tc /F4 1 Tf (. 9.1264 0 TD [(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 /F2 1 Tf 14.3462 0 0 14.3462 210.051 538.1671 Tm 0.8881 0 TD (H)Tj /F5 1 Tf /F4 1 Tf -0.781 -2.3625 TD 0.6669 0 TD /F5 1 Tf 220.959 591.807 l (,)Tj [(eo)50.1(dory)-250.1(t)0.2(he)50.2(or)50.2(em)]TJ /F2 1 Tf (,H)Tj 0.8564 0 TD /F2 5 0 R 14.3462 0 0 14.3462 190.152 289.299 Tm -16.093 -1.2052 TD 1.0955 0 TD 20.6626 0 0 20.6626 72 701.031 Tm /F4 1 Tf 0 Tc [(\). [(any)-349.9(family)]TJ /F2 1 Tf 0.0001 Tc That is, coX:= X … endstream (S,)Tj ()Tj /F5 1 Tf ()Tj -21.5619 -1.2052 TD /F4 1 Tf (\(a\))Tj /F4 1 Tf 20.6626 0 0 20.6626 347.589 529.6981 Tm 0 Tc [(�nite)-366.3(set)-365.9(of)-366.3(cardinalit)26.1(y)]TJ /F4 1 Tf 1.1369 0 TD ()Tj /F6 9 0 R 14.3462 0 0 14.3462 340.056 265.683 Tm /F3 6 0 R 20.6626 0 0 20.6626 140.004 436.3051 Tm ()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 /F2 1 Tf 0 g endobj 0.5001 0 TD [(The)-204.6(notation)-204.7([)]TJ [(Colorful)-250(Car)50.1(a)-0.1(th)24.8(�)]TJ [(bination)-393.3(o)0(f)]TJ 0 Tc 20.6626 0 0 20.6626 300.582 677.28 Tm /F6 1 Tf /F7 1 Tf )]TJ /F5 1 Tf )-405.5(let)-299.2(us)-299.2(recall)-299.2(some)-299.3(basic)]TJ (|)Tj 20.3985 0 TD /F2 1 Tf /F5 1 Tf -4.2496 -1.2052 TD /F2 1 Tf 8.1141 0 TD 20.6626 0 0 20.6626 523.467 677.28 Tm 387.657 636.416 l 9.8368 0 TD /F2 1 Tf /F4 1 Tf >> (S)Tj 6.6218 0 TD (\()Tj /F7 1 Tf /F3 1 Tf /F7 10 0 R 0.2226 Tc /Length 2115 (i)Tj /F4 1 Tf /F2 1 Tf )Tj 0.0001 Tc ()Tj endstream /F10 1 Tf /F3 1 Tf 4.8503 0 TD ()Tj 20.6626 0 0 20.6626 443.286 590.4661 Tm 220.959 620.154 m 0.3541 0 TD (. -19.1628 -1.2057 TD [(,)-349.8(s)0.2(o)-350.2(t)0.1(hat)]TJ (\))Tj (+1)Tj 0 Tc 1.0903 0 TD -19.3219 -1.2052 TD (H)Tj ()Tj /Font << /F4 1 Tf ()Tj /F3 1 Tf [(form)-280.7(de�ning)]TJ 0.6991 0 TD 20.6626 0 0 20.6626 182.34 541.272 Tm /F3 1 Tf endstream [(p)50(o)-0.1(sitive)]TJ 414.25 597.477 l /F2 1 Tf ()Tj (+)Tj >> /F4 1 Tf [(ane)-197.2(geometry:)]TJ [(eo)50.1(dory�s)-350(T)0.1(he)50.2(or)50.2(em)]TJ (H)Tj /F4 1 Tf ()Tj ()Tj /F2 1 Tf (\()Tj /F5 1 Tf endstream >> (\))Tj ({)Tj ET (\()Tj /F4 1 Tf 0.849 0 TD 0.5893 0 TD 0.0001 Tc The convex hull of a set Sis the set of all convex combinations of points in S. A justi cation of why we penalize the ‘1-norm to promote sparse structure is that the ‘1- 20.6626 0 0 20.6626 195.444 292.4041 Tm 0 Tc (H)Tj (})Tj /F2 1 Tf /F2 1 Tf /F7 1 Tf [(ma)-52.2(jor)-422.8(r)0.1(ole)-422.9(i)0.1(n)-423.4(c)0.1(on)26.1(v)26.2(e)0.1(x)-422.5(g)0(eometry)-422.9(and)-422.9(top)-26.1(o)0(logy)-422.9(\(they)-422.9(are)]TJ [(Figure)-326.8(3.1:)-435.8(\(a\))-327(A)-326(con)26.7(v)27.4(ex)-327.2(set;)-325.8(\(b\))-326.2(A)-326.8(noncon)26.7(v)27.4(e)0(x)-326.5(s)-0.1(et)]TJ 9.0336 0 TD -0.0001 Tc (E,)Tj (,)Tj 2.025 0 TD -20.6884 -1.2052 TD /F2 1 Tf (S)Tj 0.2781 Tc 20.6626 0 0 20.6626 124.938 436.3051 Tm 1.2087 0 TD /F5 1 Tf [(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? (})Tj 0 Tc (i)Tj 0 Tc 0.514 0 TD 0 Tc (. /F4 1 Tf 112.707 597.477 m 14.3462 0 0 14.3462 305.712 254.973 Tm 0 Tc (101)Tj (q)Tj 0.3541 0 TD (i)Tj endobj -2.3744 -5.9277 TD [(EODOR)81.5(Y)0(�S)-326.3(THEOREM)]TJ 0.5894 0 TD -21.7439 -2.5664 TD (a)Tj 0.6608 0 TD 0 Tc /F4 1 Tf endobj 1.0559 0 TD 0 -1.2057 TD 0 -3.184 TD 5.9251 0.7501 TD 414.25 625.823 l 20.6626 0 0 20.6626 421.299 541.272 Tm 329.211 597.477 m 0.2223 Tc 0 0 1 rg /F7 10 0 R -12.5597 -1.2052 TD 442.597 685.464 417.198 710.863 385.904 710.863 c [(Kr)50.2(ein)-295.4(and)-294.8(Milman)]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- (+1)Tj 0 Tc /F3 1 Tf (i)Tj (E,)Tj /F4 1 Tf 1.2549 0 TD 0.9443 0 TD /F2 1 Tf 0.6608 0 TD -21.2312 -1.2057 TD (+)Tj (i)Tj [(Con)26(v)26.1(ex)-424.8(sets)-425.1(also)-424.7(arise)-425.1(in)-425.2(terms)-424.7(o)-0.1(f)-425.1(h)26(yp)-26.2(erplanes. 1.0559 0 TD /F5 1 Tf )Tj ()Tj (I)Tj /F7 1 Tf 0.7305 0 TD -0.0001 Tc ()Tj /F2 1 Tf /F4 1 Tf /F9 1 Tf 0.6608 0 TD (\()Tj 0.9443 0 TD /F3 6 0 R /F2 1 Tf << /F5 1 Tf 0.6669 0 TD -20.7745 -1.2057 TD 20.6626 0 0 20.6626 136.521 468.894 Tm /Font << 0 -1.2057 TD /F4 1 Tf ET /F2 1 Tf 5.0201 0 TD /F2 1 Tf 0.6699 0 TD 27 0 obj %PDF-1.2 [(\)\))-375(and)-375.1(called)-375.5(the)]TJ [(The)-263(f)0.1(ollo)26.2(wing)-263(tec)26.2(hnical)-262.9(\()0.1(and)-263.1(dull!\))-393.2(lemma)-263(pla)26.2(y)0(s)-263(a)-263(crucial)]TJ /F2 1 Tf 1.7005 0 TD /F5 1 Tf -22.0407 -1.2052 TD ()Tj 1.1238 0 TD (v)Tj 4.6 Convex Direction: Clearly every point in the convex set (shown in blue) can be the vertex for a ray with direction [1;0]T contained entirely in the convex set. (A)Tj /F3 1 Tf /F2 1 Tf >> 14.3462 0 0 14.3462 170.163 330.0511 Tm 0 Tc 14.7128 0 TD (\()Tj f -21.1681 -1.2057 TD 0.9592 0 TD 3.5383 0 TD 0.6669 0 TD (E,)Tj 0.2781 Tc (94)Tj /F2 1 Tf 0.0001 Tc 0.0001 Tc 0.0001 Tc 0.0001 Tc (I)Tj (i)Tj /F5 1 Tf ()Tj 391.038 705.193 l /F2 1 Tf 20.6626 0 0 20.6626 177.273 333.1561 Tm /F5 1 Tf /F3 1 Tf 0.0001 Tc [(. 3.344 0 TD stream (H)Tj -14.333 -1.2052 TD endstream 0.1237 -0.7932 TD /F2 1 Tf 0.0001 Tc 1.0175 0 TD 14.3462 0 0 14.3462 134.721 433.2001 Tm /F5 1 Tf (I)Tj (m)Tj /F4 1 Tf 1.0559 0 TD << 0 g 0 Tc (=)Tj 1.0789 0 TD 0.2775 Tc [(it)-310(is)-310.5(enough)-309.7(to)-310.1(assume)-310.1(that)]TJ b 6.1448 0 TD [(dep)-26.1(e)0.1(nden)26.1(t)0.1(,)-301.3(and)-301.8(w)26.1(e)-301.8(use)-301.8(lemma)-301.4(3.2.1. endobj (i)Tj 0.2496 0 TD 0.3509 Tc 0.6669 0 TD for all z with kz − xk < r, we have z ∈ X Def. (i)Tj (�)Tj 0.8886 0 TD 2 0 obj 0.2777 Tc 0.0001 Tc 0 Tc 1.63 0 TD /F2 1 Tf 2.4118 0 TD /F2 1 Tf 20.6626 0 0 20.6626 282.096 267.4921 Tm (S)Tj 11.7064 0 TD /F2 1 Tf 1.8064 0 TD 0.3541 0 TD /F4 1 Tf 442.597 654.17 m 1.4579 0 TD /F8 16 0 R 14.3462 0 0 14.3462 320.94 401.988 Tm 0 Tc /F4 1 Tf /F3 1 Tf (�)Tj (H)Tj 9.9253 0 TD /F4 1 Tf (S)Tj 0 Tc 20.6626 0 0 20.6626 316.746 258.078 Tm -19.6267 -1.2052 TD 0.0001 Tc /F4 1 Tf endobj (V)Tj 7.4947 0 TD /F3 1 Tf (=0)Tj [(con)26.1(v)-13(\()]TJ [(con)26.1(v)-13(\()]TJ 14.3552 0 TD ET [(\))-240.8(and)]TJ (�)Tj (I)Tj 0.3499 Tc /F4 1 Tf (S)Tj /ProcSet [/PDF /Text ] /F9 1 Tf 6.6279 0 TD ()Tj [(W)78.7(e)-377.6(shall)-377.1(p)0(ro)26.2(v)26.2(e)-377.6(that)]TJ . 14.3462 0 0 14.3462 225.432 548.499 Tm 20.6626 0 0 20.6626 72 701.0491 Tm 0.994 0 TD (\()Tj endobj -20.7898 -1.2052 TD [(tan)26(t)-299.2(role)-299.3(in)-299.3(con)26(v)26.1(ex)-299.3(optimization. -0.0001 Tc /F7 1 Tf -5.1077 -1.7841 TD ()Tj -22.3496 -1.2052 TD (for)Tj 0 -1.2052 TD -0.0003 Tc -6.969 -1.2052 TD 20.6626 0 0 20.6626 388.278 493.7971 Tm /F1 1 Tf 345.875 611.65 m /F4 1 Tf 0.3541 0 TD [(ve)50.1(ctors)-306.9(i)-0.1(n)]TJ -17.1657 -2.941 TD (\()Tj )]TJ /F7 1 Tf 0.6991 0 TD 14.3462 0 0 14.3462 202.761 289.299 Tm /F2 1 Tf [(p)50.1(o)0(lyhe)50.2(dr)50.2(al)-350.1(c)50.2(one)]TJ 1.0554 0 TD 14.3462 0 0 14.3462 233.586 433.2001 Tm [(h)26.1(y)0(p)-26.1(erplane)]TJ [(Given)-516.7(any)-516.4(ve)50.1(ctor)-517(sp)50(ac)50.1(e,)]TJ /F9 1 Tf /F8 1 Tf 0.2836 Tc /F2 1 Tf 7.033 0 TD 3.3096 0 TD /F4 1 Tf /F4 1 Tf ()Tj >> (S)Tj /F2 1 Tf >> 0.0001 Tc /F2 1 Tf (i)Tj 379.485 636.416 m -14.8212 -2.8447 TD 2.8875 0 TD /F1 1 Tf /F4 1 Tf BT (L,)Tj /F7 1 Tf (f)Tj b BT 0.5711 0 TD 0 J 0 j 0.996 w 10 M []0 d -0.0001 Tc 14.3462 0 0 14.3462 93.807 463.0081 Tm )-762.6(CARA)81.1(TH)]TJ 0.6608 0 TD 0.6669 0 TD 20.6626 0 0 20.6626 295.929 258.078 Tm 20.6626 0 0 20.6626 170.811 468.894 Tm (\()Tj /F5 1 Tf •Convex optimization ()is a convex function, is convex set •ut “today’s problems”, and this tutorial, are non-convex •Our focus: non-convex problems that arise in machine learning Variable, in function feasible set (b)Tj endobj 0 Tc /F3 1 Tf (})Tj /F2 1 Tf 0.8563 0 TD 0 Tc -0.0001 Tc /F4 1 Tf /F4 1 Tf -18.7984 -1.2057 TD [(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 0 Tc (\()Tj (E)Tj /F8 1 Tf [(is)-301.9(con)26(v)26.1(ex. 1.386 0 TD 0.5893 0 TD 0 Tc >> /F5 1 Tf 0.3938 Tc 14.3462 0 0 14.3462 281.808 240.78 Tm /F4 1 Tf 6.5822 0 TD [(spanned)-266.1(b)26.1(y)]TJ 0 0 1 rg 14.3462 0 0 14.3462 109.458 587.3701 Tm -0.0261 Tc stream /F2 1 Tf /F3 1 Tf )-406.2(B)0.2(y)-302.3(lemma)-301.4(3.1.2,)]TJ 14.3462 0 0 14.3462 219.006 573.402 Tm 0 -1.2052 TD (\()Tj /F4 1 Tf (�)Tj /F4 1 Tf endstream /F4 1 Tf 0 -2.363 TD /F2 1 Tf (a)Tj (1)Tj 0.6608 0 TD 11.9551 0 0 11.9551 72 736.329 Tm 0 0 1 rg 1.2658 0 TD /F2 1 Tf 1.0855 0 TD 0 Tc /F2 1 Tf << /F2 1 Tf /F1 1 Tf (�s. /F5 1 Tf [(=\()277.7(1)]TJ 220.959 620.154 l [(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 (�)Tj 0 -1.2052 TD 0.0001 Tc ()Tj /F4 1 Tf 0 g /F5 1 Tf [(a,)-166.6(b)]TJ /F1 1 Tf 0 Tc <> /F4 1 Tf f 0.2775 Tc 0.8947 0 TD [(,i)366.7(f)]TJ /F8 1 Tf 7.053 0 TD )-374.1(If)-205.1(the)-205.2(t)0(heorem)-204.9(is)-205.2(false,)-224.3(there)]TJ [(is)-306.8(a)-307(c)50.2(onvex)-306.9(c)50.2(o)0(mbina-)]TJ 1.1665 0 TD 0 Tc /F2 1 Tf /F4 1 Tf /F8 1 Tf [(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 20.6626 0 0 20.6626 72 517.845 Tm [(\(2\))-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 15 0 obj 0.0001 Tc 0.0001 Tc 9.068 0 TD 0 0 1 rg 0.889 0 TD (i)Tj 0 -2.7349 TD 0 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 (\))Tj )-590.1(Giv)26.1(e)0(n)-363.4(a)-0.1(n)26(y)-362.9(set)-363.3(of)-362.8(v)26.1(ectors,)]TJ )-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 It can be proved that under mild conditions midpoint convexity implies convexity. 0.389 0 TD 20.6626 0 0 20.6626 72 499.044 Tm [(Lemma)-375.4(3.2.1)]TJ 0.3541 0 TD (? The convex hull of a set Scontains all convex combination of points in S. Intuitively, it is the smallest convex set that contains S. De nition 1.5 (Convex hull). 13.4618 0 TD /F5 1 Tf [(p)-26.2(o)-0.1(in)26(ts)-301.9(in)]TJ 0.0001 Tc /F5 8 0 R /F4 1 Tf /F8 1 Tf /F2 1 Tf BT 0.7836 0 TD /F7 1 Tf 1.494 w 5.2234 -1.7841 TD 1.6295 0 TD 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�\(�")�eHQ��9���BE��Q\�X2ʢz���r�4�����߳�Pz�7-�j. )]TJ (I)Tj >> [(,)-310.9(w)-0.1(e)-306.2(h)-26.2(ave)]TJ 12.8124 0 TD (such)Tj 0 g -9.2888 -2.3625 TD (102)Tj /F4 1 Tf /F2 1 Tf /F4 1 Tf /F2 1 Tf (E)Tj /F4 1 Tf /F4 1 Tf ET 2.5634 0 TD 0.862 0 TD 13.4618 0 TD /F9 1 Tf 0.4587 0 TD (. /F2 1 Tf (a)Tj S /F4 1 Tf /Font << 1.001 0 TD 0 Tc 9.6504 0 TD /F1 4 0 R 0.7836 0 TD 0.9974 0.7501 TD /F4 1 Tf /F8 1 Tf >> /F5 1 Tf /F7 1 Tf [(c)50.2(o)0(mbinations)]TJ /F2 1 Tf /F7 1 Tf 22 0 obj [(There)-254.8(is)-254.8(also)-254.9(a)-254.9(v)26.1(ersion)-254.5(o)-0.1(f)-255.2(T)-0.2(heorem)-254.6(3.2.2)-254.9(f)0(or)-254.8(con)26(v)26.1(ex)-254.4(cones. f (i)Tj 5.5102 0 TD -18.5359 -1.2052 TD 391.038 676.846 l (\))Tj (is)Tj 0 -2.3625 TD << /F4 1 Tf 0 Tc 0.0001 Tc 0.2778 Tc /F2 1 Tf (\))Tj 0.0001 Tc /F4 1 Tf (i)Tj /F4 1 Tf /F4 1 Tf 1.8059 0 TD 0.5001 0 TD 9.6003 0 TD (|)Tj /F4 1 Tf 14.3462 0 0 14.3462 325.017 573.402 Tm 0.7836 0 TD 6.6699 0.2529 TD 0.7392 0 TD /F3 1 Tf ()Tj 427.245 613.792 426.308 612.855 426.308 611.7 c 0 Tc /F4 1 Tf 7.6254 0 TD -21.4158 -1.2052 TD [(,)-427.2(t)0(here)-402.1(is)-402.5(a)]TJ -10.1165 -1.2057 TD /F4 1 Tf 0 Tc ({)Tj (�)Tj 0 -1.2052 TD /F2 1 Tf /F5 1 Tf /F2 1 Tf 0.72 0 TD /F3 1 Tf /F2 1 Tf (> /F4 1 Tf 0.333 Tc 0 Tc 20.6626 0 0 20.6626 258.93 195.0601 Tm (v)Tj 112.707 625.823 m /F7 10 0 R BT /F4 7 0 R /ExtGState << 226.093 654.17 l 14.3462 0 0 14.3462 356.058 239.493 Tm (i)Tj /F4 1 Tf [(de�nitions)-301.8(ab)-26.1(out)-301.8(cones. /F5 1 Tf /F2 1 Tf -8.4369 -1.2052 TD BT 0 Tc S /F2 1 Tf /F2 1 Tf 14.3462 0 0 14.3462 369.252 261.6151 Tm [(of)-400.3(p)-26.2(o)-0.1(in)26(ts)-399.9(in)]TJ /F4 1 Tf /Length 3049 0.2779 Tc /F5 1 Tf /GS1 gs /F2 1 Tf ()Tj >> (>)Tj (q)Tj 0 Tw (+1)Tj 0 Tc (If)Tj (E)Tj (i)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 /F2 1 Tf 0 Tc (i)Tj 0 Tc 8.4608 0 TD (i)Tj 0.0001 Tc 0.6943 0 TD (ane)Tj (|)Tj /F2 1 Tf 0.5798 0 TD 11.1776 0 TD 11.9551 0 0 11.9551 72 736.329 Tm (i)Tj 0 0 1 rg [(for)-349.8(every)]TJ 46 0 obj 20.6626 0 0 20.6626 323.226 195.0601 Tm ()Tj 14.3462 0 0 14.3462 253.656 264.3961 Tm (\))Tj (\). (C)Tj In our 0 Tc In Euclidean space, a region is a convex set if the following is true. /GS1 11 0 R (,...,a)Tj (:)Tj )Tj 1.9361 0 TD /F4 1 Tf Cone: a cone is a set that is closed under multiplication by positive scalars, i.e. 0.2779 Tc (a)Tj [(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 4.4007 0 TD 1.0554 0 TD 357.557 597.477 l 0.5763 0 TD 0.0001 Tc /F4 1 Tf 14.3462 0 0 14.3462 161.964 548.499 Tm (I)Tj -0.0001 Tc 1.0528 0 TD [(,)-349.8(and)]TJ 1 i 0.967 0 TD 0.72 0 TD 0.788 0 TD 0 Tc [(has)-224.2(dimension)]TJ 14.3462 0 0 14.3462 406.674 264.3961 Tm 414.25 597.477 l BT (ing)Tj 17.5298 0 TD /F4 1 Tf 2.0838 0 TD /F2 1 Tf /F2 1 Tf /F2 1 Tf 14.3462 0 0 14.3462 216.234 261.6151 Tm 0 Tc /F4 1 Tf 11.9551 0 0 11.9551 72 736.329 Tm 0 Tc /F4 1 Tf [(T)-50.1(h)-50.2(eo)-50.2(re)-50.1(m)]TJ ()Tj -0.0001 Tc [(. /F11 25 0 R 226.093 654.17 l (f)Tj (\))Tj 0 Tc 2.0207 0 TD /F2 1 Tf 10.2528 0 TD [(Similarly)78.3(,)-424.6(t)0(he)-400.3(con)26(v)26.1(ex)-399.9(h)26(u)-0.1(ll)-400.2(of)-400.3(a)-399.9(set)]TJ /F5 1 Tf 5.9074 0 TD 0 Tc 0 Tc -18.4184 -2.3625 TD 14.3462 0 0 14.3462 89.937 540.5161 Tm ()Tj BT /F2 1 Tf 442.597 597.477 l (and)Tj 3.9246 0 TD Convex hull of a set: set of all convex combinations of points in the set. /F4 1 Tf [(=K)277.5(e)277.7(r)]TJ 357.557 625.823 l 0.3541 0 TD (\()Tj (\()Tj 0.0001 Tc 138.105 710.863 112.707 685.464 112.707 654.17 c /F4 1 Tf (i)Tj 6.5129 0 TD 0.9443 0 TD /F2 1 Tf 14.3462 0 0 14.3462 303.831 638.9041 Tm /F4 1 Tf [(,)-301.7(t)0.1(hat)-301.8(is,)]TJ 2.1483 0 TD /F2 1 Tf 0.75 g 0.1111 0 TD 0 Tc /F2 1 Tf /F5 1 Tf 21.1364 0 TD [(CHAPTER)-327.3(3. (93)Tj 1.1604 0 TD 0 0 1 rg /F2 1 Tf 1.1194 0 TD [(com)26(b)0(inations)-301.3(of)]TJ (\()Tj (�)Tj (0)Tj /F2 1 Tf 11.9551 0 0 11.9551 378.099 572.1901 Tm 112.707 654.17 l 0 Tc 0.0002 Tc /F4 1 Tf 0.3541 0 TD Convex combination and convex hull convex combination of x1,. )]TJ /F7 1 Tf /F9 20 0 R 20.6626 0 0 20.6626 351.477 268.7791 Tm stream stream This provides a bridge between a geometric approach and an analytical approach in dealing with convex functions. >> 11.7569 0 TD 0.8407 0 TD [(F)78.6(o)0(r)-327.5(t)0.1(his)-327.5(reason,)-333.9(w)26.1(e)-327(will)-327.4(also)-327.5(sa)26.2(y)-327.5(t)0.1(hat)]TJ 0 Tc [(,s)315.1(p)365(a)314.9(n)314.8(n)314.9(e)365.1(d)8.3(b)315(y)]TJ 14.3462 0 0 14.3462 358.362 404.769 Tm 0 Tc 0 g Formally, a set CˆRN is said to be convex if for any x 1 and x 2 in Cthe point x 1 + (1 )x 2 2Cfor any 2[0;1]. 0.0001 Tc /F3 1 Tf ({)Tj /GS1 11 0 R /F7 1 Tf /F4 1 Tf 0.514 0 TD 0.3338 0 TD (\))Tj 14.3462 0 0 14.3462 210.051 660.4141 Tm /F4 1 Tf -22.3781 -1.7837 TD [(is)-350.1(also)-349.8(c)50.1(o)-0.1(mp)50(act. [(c)50.1(onvex)-350.2(p)50(o)-0.1(lytop)50(e)0(s)]TJ 20.6626 0 0 20.6626 199.431 541.272 Tm 414.25 625.823 l ()Tj [(,)-448.7(for)]TJ ET [(of)-350.2(p)50.1(oints)-349.5(i)0(n)]TJ /F3 1 Tf 0.6608 0 TD 0 Tc (\))Tj 8.8337 0 TD [(,o)349.8(f)]TJ 14.3462 0 0 14.3462 78.633 411.6901 Tm If every point on that segment is inside the region, then the region is convex. -15.875 -1.2052 TD (=1)Tj -0.0001 Tc (,)Tj 0.0001 Tc -16.2673 -1.2057 TD stream [(Ho)26.1(w)26(e)0(v)26.1(er,)-395(the)-376.8(set)]TJ /F4 1 Tf (If)Tj 1.1604 0 TD /F2 1 Tf BT 3.9573 0 TD [(Given)-429.6(an)-429.2(ane)-429.4(sp)50(ac)50.1(e)]TJ /F5 1 Tf 0.6608 0 TD (H)Tj CONVEX SET RECONSTRUCTION 415 quantities precisely. (+)Tj << (\))Tj (i)Tj /F8 16 0 R convex optimization, i.e., to develop the skills and background needed to recognize, formulate, and solve convex optimization problems. (\(i.e.,)Tj 0.2617 Tc f /F4 1 Tf /F4 1 Tf /F3 1 Tf /GS1 gs (+)Tj /F5 1 Tf 20.6626 0 0 20.6626 199.062 590.4661 Tm ()Tj [(S,)-384.2()]TJ /F4 1 Tf /F7 1 Tf 0 g ()Tj 0.1667 Tc /F4 1 Tf Daniel P. Palomar 6 0.3338 0 TD -0.0002 Tc 0.3541 0 TD (\))Tj [(state,)-270.9(but)-263.1(they)-263.6(are)-263.1(d)-0.1(eep,)-270.9(and)-263.2(their)-263.1(pro)-26.2(o)-0.1(f,)-270.9(although)-263.2(rather)]TJ 1.0014 -1.7841 TD -18.5395 -1.2052 TD /F1 1 Tf (E,)Tj 0 Tc 7.3645 0 TD 0 Tw T* -22.3501 -1.2052 TD (i)Tj (f)Tj 0.3391 Tc /F5 1 Tf -13.3009 -3.3269 TD 1.6025 0 TD 0 Tc 20.6626 0 0 20.6626 249.741 576.498 Tm 0 Tc /F5 1 Tf [(a)-340.1(c)0.1(on)26.1(v)26.2(e)0.1(x)-339.7(set)-340.1(whic)26.2(h)-339.7(i)0.1(s)-340.1(a)0(lso)-340.1(compact)-339.7(i)0.1(s)-340.1(t)0.1(he)-340.1(con)26.1(v)26.2(ex)-339.7(h)26.1(u)0(ll)-340(of)]TJ 0.3541 0 TD 0.4587 0 TD ()Tj /F5 1 Tf /F2 1 Tf 0.5893 0 TD (I)Tj (i)Tj /F7 1 Tf 20.6626 0 0 20.6626 365.103 590.4661 Tm 0 Tc (b)Tj 0 -2.3625 TD -0.0001 Tc /F2 1 Tf 0 Tc [(CHAPTER)-327.3(3. 3.3313 0 TD 1.6291 0 TD -19.8488 -1.2052 TD 0.446 Tc (. 14.3462 0 0 14.3462 344.844 660.4141 Tm /F2 1 Tf (I)Tj 9.3037 0 TD 0 Tc /ExtGState << 0.4164 0 TD /F5 1 Tf 0.3541 0 TD 442.597 654.17 m /F5 1 Tf /F9 20 0 R (I)Tj 0.3541 0 TD 0.2503 Tc /F4 1 Tf 1.1168 0 TD 4.0253 0 TD (E)Tj /F5 1 Tf (=0)Tj 0.0001 Tc (E)Tj 5.139 0 TD 0.6904 0 TD 20.6626 0 0 20.6626 333.045 541.272 Tm /F4 1 Tf -14.6853 -1.2052 TD (of)Tj /F4 1 Tf 0.9861 0 TD (S)Tj [(0,)-423.2(then)]TJ [(space)-301.8(o)0(f)]TJ 20.6626 0 0 20.6626 336.042 576.498 Tm 0.3541 0 TD /F2 1 Tf /F7 1 Tf )Tj /F4 1 Tf 0.0001 Tc 0.6669 0 TD /F2 1 Tf 0.4587 0 TD 5.9912 0 TD , a straight line segment can be mathematically demanding, especially for the reader primarily... 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