/F3 1 Tf [(cannot)-301.8(b)-26.1(e)-301.3(expressed)-301.8(as)-301.8(an)26.1(y)-301.8(c)0.1(on)26.1(v)26.2(e)0.1(x)-301.8(c)0.1(om)26(bination)]TJ 0 Tc 3.1. )-762.6(CARA)81.1(TH)]TJ -18.5408 -1.2057 TD [(,)-306.1(i)0.1(s)-305.7(t)0.1(he)-305.7(dimension)]TJ /F5 1 Tf 0.3938 Tc 0.2783 Tc 42 0 obj (i)Tj /F2 1 Tf (E)Tj [(CHAPTER)-327.3(3. -20.7745 -1.2057 TD [(3.2. /F5 8 0 R 0 Tc ()Tj (m)Tj 0.7366 0 TD 442.597 597.477 m /F2 1 Tf /F2 1 Tf 0.6773 0 TD 5.2234 -1.7841 TD 20.6626 0 0 20.6626 221.58 541.272 Tm (a)Tj /F2 1 Tf (101)Tj 427.245 613.792 426.308 612.855 426.308 611.7 c 379.786 629.139 m 0 Tc ()Tj 0 Tc -0.0001 Tc 0.0001 Tc 0 Tc (})Tj 0 Tc /ExtGState << [7], Given r points u1, ..., ur in a convex set S, and r 24.7871 0 0 24.7871 72 624.873 Tm 0.3541 0 TD /F7 1 Tf 0.0001 Tc 0.72 0 TD )Tj 0.5101 0 TD A set that is not convex is called a non-convex set. 0.0001 Tc /F4 1 Tf 0.862 0 TD 0 -1.2052 TD 0.0001 Tc /F6 9 0 R [(1o)393.7(ft)393.8(h)393.7(e)]TJ 0 Tc [(Theorem)-375.9(3.2.2)]TJ We can inscribe a rectangle r in C such that a homothetic copy R of r is circumscribed about C. The positive homothety ratio is at most 2 and:[10], The set 20.6626 0 0 20.6626 404.523 652.368 Tm 0 Tc 1 0 0 RG /F4 1 Tf 0.994 0 TD 10.0402 0 TD 442.597 654.17 l (i)Tj /Length 2115 1.6465 0 TD /F6 1 Tf 0.5893 0 TD (H)Tj 6.6161 0 TD 0.5367 Tc BT >> 1.9361 0 TD 1.2153 0 TD (S)Tj 0.0041 Tc /F2 1 Tf 0 Tc /F7 1 Tf /F4 1 Tf Let Y ⊆ X. 0 Tc [(=\()277.7(1)]TJ 0.3541 0 TD [(,i)354.9(s)10.4(a)]TJ 0 0 1 rg (I,)Tj /F1 1 Tf 0 Tc /GS1 gs /F2 1 Tf 14.3462 0 0 14.3462 216.234 261.6151 Tm 1.143 0 TD (H)Tj ({)Tj 0.389 0 TD (c)Tj BT (S)Tj /F4 7 0 R 0 Tc 1.0437 0 TD /F5 1 Tf /F2 1 Tf S /F4 1 Tf -18.6998 -1.2057 TD 0.5798 0 TD 0.0001 Tc [(p)50(oints,)]TJ 20.6626 0 0 20.6626 72 701.031 Tm 1.4566 0 TD )Tj C, a vector. [(,w)350(i)349.8(t)350(h)]TJ 1.3691 0 TD (=)Tj 0.3541 0 TD 14.3462 0 0 14.3462 358.362 404.769 Tm /F2 1 Tf (of)Tj [(c)50.2(o)0(mbinations)]TJ /F2 1 Tf 14.3462 0 0 14.3462 210.051 538.1671 Tm 0.0001 Tc 13.9283 0 TD 0 -1.2052 TD >> 0 Tc /Font << /F5 1 Tf /F3 1 Tf )-406.2(B)0.2(y)-302.3(lemma)-301.4(3.1.2,)]TJ 0.3337 0 TD -18.0969 -2.3625 TD Linear algebra proof that this set is convex mathematics stack. 0 g -1.4409 3.3061 TD 0 Tc 20.6626 0 0 20.6626 333.045 541.272 Tm (i)Tj 0.3541 0 TD 1.6291 0 TD 0 Tc /F4 1 Tf /ProcSet [/PDF /Text ] )-761.6(BASIC)-326.4(P)0(R)27.3(O)-0.3(PER)81.5(TIES)-326.3(OF)-326.1(CONVEX)-326.7(SETS)]TJ (i)Tj 0 Tc [(\)i)283.7(st)283.6(h)283.5(e)]TJ /F2 1 Tf /ProcSet [/PDF /Text ] 220.959 662.673 m 9.8368 0 TD 0 Tc /ProcSet [/PDF /Text ] = ()Tj 0 Tc /F4 1 Tf /F4 1 Tf 0.6608 0 TD 20.6626 0 0 20.6626 523.467 677.28 Tm 14.3462 0 0 14.3462 340.056 265.683 Tm T* [(,)-349.8(s)0.2(o)-350.2(t)0.1(hat)]TJ /ExtGState << 14.3462 0 0 14.3462 303.831 516.657 Tm >> endobj [(CHAPTER)-327.3(3. )Tj [(has)-393.7(dimension)]TJ (+1)Tj /F2 1 Tf 0 Tc 1.8059 0 TD (I)Tj (1)Tj ()Tj (H)Tj /F2 1 Tf /F4 1 Tf [(,)-558(o)0(f)-516.6(di-)]TJ ()Tj 1.782 0 TD (X)Tj /F7 1 Tf 0.3541 0 TD (0)Tj /F1 1 Tf -4.4777 -2.2615 TD /F4 1 Tf /F10 1 Tf [(CHAPTER)-327.3(3. (f)Tj /F2 1 Tf /F4 1 Tf 3.4093 0 TD (\))Tj 20.6626 0 0 20.6626 255.204 541.272 Tm (“nite)Tj (I)Tj /F4 1 Tf /F4 1 Tf (R)Tj 0 -1.2052 TD ()Tj [(,)-315.4(t)0.2(hat)-306.9(is,)]TJ (|)Tj 0.6608 0 TD -10.1165 -1.2057 TD /F4 1 Tf (])Tj /F1 1 Tf (97)Tj 0.5558 0 TD (1)Tj /F4 1 Tf 0 Tc 0 Tc )Tj 1.8726 0 TD /F5 1 Tf [(,)-421.7(for)-406.7(any)-406.9(\(nonvoid\))-406.6(family)]TJ S (i)Tj [(])-301.7(o)0(r)-301.8(L)0.2(ang)-301.8([)]TJ (S)Tj 0.1111 0 TD )-405.5(let)-299.2(us)-299.2(recall)-299.2(some)-299.3(basic)]TJ (\()Tj 1.065 0 TD >> 0.632 0 TD (i)Tj ()Tj /F2 1 Tf (:)Tj 0 -1.2057 TD (? 0.9861 0 TD /F5 1 Tf /F7 1 Tf << 14.3462 0 0 14.3462 325.017 573.402 Tm 14.3462 0 0 14.3462 389.178 649.272 Tm 31.1377 0 TD If A or B is locally compact then A − B is closed. Rawlins G.J.E. 31 0 obj /F4 1 Tf /F2 1 Tf /F3 1 Tf -17.1657 -2.941 TD 0.1667 Tc /F2 1 Tf 20.6626 0 0 20.6626 255.204 663.519 Tm 14.3462 0 0 14.3462 440.334 265.683 Tm 0 Tc /F4 1 Tf ()Tj ()Tj /F5 1 Tf 1.1604 0 TD 14.3462 0 0 14.3462 320.94 401.988 Tm (a)Tj [(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. (i)Tj /F2 1 Tf 14.3462 0 0 14.3462 206.883 623.217 Tm -0.0002 Tc (q)Tj [(line)50.2(ar)-365.8(c)50.2(o)0(mbinations)]TJ (a)Tj /F4 1 Tf 0.0001 Tc (+)Tj /F4 1 Tf /F2 1 Tf (\()Tj [(has)-330.5(“nite)-330.5(supp)-26.1(ort)-330.1(\()0.1(all)]TJ (,)Tj 7.1828 0 TD 0.8564 0 TD (f)Tj S 0 Tc 0.3338 0 TD 0 Tc ()Tj 0.5893 0 TD (a)Tj 11.9551 0 0 11.9551 72 736.329 Tm /F2 1 Tf 1.1451 0 TD 4.7087 0 TD [(tan)26(t)-299.2(role)-299.3(in)-299.3(con)26(v)26.1(ex)-299.3(optimization. (S)Tj 3 0 obj (ar´)Tj /F3 1 Tf is a linear subspace. /F3 1 Tf -21.7619 -1.2057 TD (+1)Tj /F4 1 Tf /F1 1 Tf /F5 1 Tf (mension)Tj In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points, it contains the whole line segment that joins them. 0.3541 0 TD 0 Tc /F3 1 Tf [(\),)-423.8(but)-399(if)]TJ /F3 1 Tf S /F5 1 Tf 1.2549 0 TD (called)Tj 442.597 654.17 m 0 Tc (a)Tj 0.5893 0 TD This result holds more generally for each finite collection of non-empty sets: In mathematical terminology, the operations of Minkowski summation and of forming convex hulls are commuting operations. 0 Tc /F2 1 Tf K D 14.3462 0 0 14.3462 196.695 403.1671 Tm /F3 1 Tf ()Tj 5.139 0 TD 3.3313 0 TD ()Tj d. is a. direction of recession. (\()Tj (c)Tj (. >> 0.7919 0 TD (i)Tj /F4 1 Tf 19.3423 0 TD [(the)-324(c)50.1(onvex)-323.6(hul)-50.1(l)]TJ (S)Tj (. (E)Tj 387.657 636.416 l [(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. 2.2019 0 TD )-762.5(CONVEX)-326(SETS)]TJ -22.0456 -2.3625 TD /F5 1 Tf [(,)-493.6(let)]TJ /F2 1 Tf 0.3509 Tc 1.8064 0 TD 20.6626 0 0 20.6626 237.609 626.313 Tm 14.3462 0 0 14.3462 339.822 487.911 Tm 0 Tc /Length 4531 (f)Tj The convex-hull operator Conv() has the characteristic properties of a hull operator: 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. ()Tj [(,)-546.8(for)-507.4(any)-507.7(se)50.1(quenc)50.1(e)-508.3(of)]TJ /F2 1 Tf [(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 0.5554 0 TD 0.3038 Tc 0 g [(ve)50.1(ctors)-306.9(i)-0.1(n)]TJ 0.9361 0 TD (and)Tj [(is)-344.5(closed)-344.6(under)]TJ [(\(1\))-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 0.5893 0 TD 1.2113 0.95 TD /F5 8 0 R /F4 1 Tf /F2 1 Tf /F5 1 Tf ( )Tj /F5 1 Tf 0 Tw 14.3462 0 0 14.3462 160.092 465.7891 Tm Then x ∈ A because A is convex, and similarly, x ∈ B because B is convex. {\displaystyle \operatorname {rec} S} 14.3462 0 0 14.3462 131.229 465.7891 Tm ({)Tj (Š)Tj >> 2 /F3 1 Tf /F2 1 Tf ()Tj ()Tj /F3 1 Tf S /F2 1 Tf 0 Tc (})Tj 0 Tc /F2 1 Tf 0 0 1 rg 0.0001 Tc (i)Tj 2 6.5129 0 TD and satisfying /ProcSet [/PDF /Text ] /F2 1 Tf >> [(consists)-322.3(of)]TJ 0 Tc /F1 1 Tf 0.7379 0 TD /F4 1 Tf /F4 1 Tf 0.6608 0 TD /F5 1 Tf 1.0358 0 TD /F5 1 Tf (\))Tj (i)Tj /F3 1 Tf 6.5822 0 TD (f)Tj Let A and B be non-empty, closed, and convex subsets of a locally convex topological vector space such that /F2 1 Tf /GS1 11 0 R /F5 1 Tf (i)Tj 0.0001 Tc /F4 1 Tf 3.9516 0 TD Let C be a convex body in the plane (a convex set whose interior is non-empty). /F5 1 Tf 1.0611 0 TD 0 Tc 2.8875 0 TD 0.0001 Tc (Š)Tj [(It)-297.1(is)-296.6(ob)26(vious)-297.1(that)-296.6(the)-296.7(i)0(n)26(tersection)-297.1(o)-0.1(f)-297(a)-0.1(n)26(y)-296.7(family)-296.7(\(“nite)-297.1(or)]TJ (\). /F2 1 Tf (E)Tj >> 0.3541 0 TD /F5 1 Tf 138.105 710.863 112.707 685.464 112.707 654.17 c (b)Tj 2.2328 0 TD -18.1958 -3.7215 TD (for)Tj For an alternative definition of abstract convexity, more suited to discrete geometry, see the convex geometries associated with antimatroids. (m)Tj /F2 1 Tf 22 0 obj /F2 1 Tf /F4 1 Tf /F3 1 Tf (=)Tj 0.0001 Tc 20.6626 0 0 20.6626 347.589 529.6981 Tm /F2 1 Tf [(,)-375.8(recall)-361.5(that)-361.5(a)-361.1(s)0(ubset)]TJ /F4 1 Tf 0 Tc (\()Tj (S)Tj /F5 1 Tf 0.585 0 TD T* (\))Tj 1.0855 0 TD 4.7292 0 TD /F5 1 Tf /F3 1 Tf -0.0001 Tc /F2 1 Tf 0.6608 0 TD /F3 1 Tf (A)Tj 0.3338 0 TD 0.3042 Tc 414.25 625.823 l 1.0689 0 TD The following famous theorem, proved by Dieudonné in 1966, gives a sufficient condition for the difference of two closed convex subsets to be closed. 14.3462 0 0 14.3462 471.411 515.6041 Tm 14.3462 0 0 14.3462 153.135 638.9041 Tm 379.485 636.416 m 0.3541 0 TD 2 0 obj 0.3541 0 TD -0.0001 Tc )]TJ (. 20.6626 0 0 20.6626 72 702.183 Tm /F2 1 Tf ()Tj /F3 6 0 R 0 Tc 226.093 654.17 l 0.6608 0 TD /F2 1 Tf /F4 1 Tf /F2 1 Tf /F3 1 Tf (a)Tj {\displaystyle 0\in X} /F4 1 Tf (Š)Tj (S)Tj (. 0.0001 Tc Note that if S is closed and convex then (=0)Tj (. [(\))-240.8(and)]TJ ({)Tj 0 -1.2052 TD 0.849 0 TD (j)Tj (\()Tj 4.7126 0 TD 0 Tc 0 0 1 rg (+1)Tj 1.2209 0 TD 0.6608 0 TD 329.211 597.477 l /Font << /F2 1 Tf [(of)-251.6(a)-251.7(s)0.1(ubset,)]TJ (|)Tj /F5 1 Tf 0.6608 0 TD 0.0001 Tc ()Tj (S)Tj >> 0 Tc /F2 1 Tf (\). (E)Tj /F2 1 Tf Convex Optimization - Polyhedral Set - A set in $\mathbb{R}^n$ is said to be polyhedral if it is the intersection of a finite number of closed half spaces, i.e., 391.038 705.193 l /F2 1 Tf /F4 1 Tf 20.6626 0 0 20.6626 258.93 195.0601 Tm The branch of mathematics devoted to the study of properties of convex sets and convex functions is called convex analysis. 0 -1.2057 TD [(is)-251.8(the)-251.8(s)0(mallest)-251.3(ane)-251.8(set)-251.8(con)26(t)0(ain-)]TJ ∩ /F2 1 Tf /F9 1 Tf (i)Tj 0.6669 0 TD [(is)-351.1(a)]TJ /GS1 gs T* (I)Tj /F3 1 Tf /F3 1 Tf /F1 4 0 R (j)Tj ()Tj 1.9305 0 TD 0.0001 Tc /F8 16 0 R 1.0554 0 TD /F2 1 Tf /F2 5 0 R 0 Tc [(\(close)50.1(d\))-350.5(half)-349.9(s)0.1(p)50(a)-0.1(c)50.1(e)0(s)-350.5(a)-0.1(sso)50(ciate)50.1(d)-350.3(with)]TJ 1.1194 0 TD /F3 1 Tf 2.7455 0 TD /F2 1 Tf 0.6608 0 TD (a)Tj ET >> 7.3645 0 TD -20.6834 -1.2057 TD 14.3462 0 0 14.3462 89.937 540.5161 Tm /F4 1 Tf 11.9551 0 0 11.9551 161.928 572.1901 Tm The theorem simpli es many basic proofs in convex analysis but it does not usually make veri cation of convexity that much easier as the condition needs to hold for all lines (and we have in nitely many). /F2 1 Tf BT 0 g ()Tj 0.6669 0 TD C. ... all level sets are compact. 226.093 685.464 200.694 710.863 169.4 710.863 c 0.0001 Tc 0 g Now let R±∞ = R ∪{−∞,∞} be the ordered set of (doubly) extended reals.Foreach elementr ofthe setR+ of strictly positive reals, a binary operationr will be deﬁned /F4 1 Tf (with)Tj /F5 1 Tf [(In)-304.5(view)-305(of)-304.3(lemma)-304.4(3.1.2,)-305.2(it)-304.4(is)-304.8(ob)26.1(vious)-304.4(that)-304.4(an)26.1(y)-304.9(a)0(ne)-304.4(s)0.1(ub-)]TJ 0.8341 0 TD /F4 1 Tf (\))Tj nonnegative numbers λ1, ..., λr such that λ1 + ... + λr = 1, the affine combination. /F2 1 Tf (i)Tj /F4 1 Tf 0.0001 Tc 14.3462 0 0 14.3462 325.719 649.272 Tm /F3 1 Tf (i)Tj (c)Tj (S)Tj 0.849 0 TD 1.1769 0 TD [(called)-301.9(a)]TJ (j)Tj 0 Tc (a)Tj (a)Tj /F2 5 0 R 1.369 0 TD [(is)-301.9(allo)26.1(w)26(e)0(d\). [(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 {\displaystyle \operatorname {rec} S=\bigcap _{t>0}t(S-s_{0})} /F4 1 Tf 0 Tc 0.5893 0 TD /F4 1 Tf (\). 14.3462 0 0 14.3462 338.004 254.973 Tm 0.8881 0 TD /F7 1 Tf /F4 1 Tf ()Tj 0.6608 0 TD /F2 1 Tf We say a set Cis convex if for any two points x;y2C, the line segment (1 )x+ y; 2[0;1]; lies in C. The emptyset is also regarded as convex. [(EODOR)81.5(Y)0(’S)-326.3(THEOREM)]TJ (+)Tj /F3 1 Tf 0 -2.3625 TD /F6 1 Tf /F3 1 Tf 0.3541 0 TD endstream 5.0201 0 TD 11.9551 0 0 11.9551 72 736.329 Tm /F3 6 0 R /GS1 11 0 R can also be parametrized by its width (the smallest distance between any two different parallel support hyperplanes), perimeter and area.[11][12]. [(or)-301.8(a)-59.1(\()]TJ /F5 1 Tf /F2 1 Tf 38.1668 0 TD /F9 1 Tf Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets. 0.0527 -0.7187 TD -0.0002 Tc 0.3541 0 TD 20.3985 0 TD (f)Tj ⋂ (I)Tj /Font << /F5 1 Tf (S)Tj 20.6626 0 0 20.6626 72 701.0491 Tm -0.0001 Tc 0.4164 0 TD (R)Tj 0 Tc << = 0.5001 0 TD endobj [(c)50.2(onvex)-390.6(c)50.2(one)]TJ (})Tj 0.6669 0 TD /F2 1 Tf 357.557 597.477 m 1.0001 0 TD endstream (\()Tj 0 Tc /F3 1 Tf [(. 0 -1.2052 TD 0 0 1 rg [(DeŞnition)-375.6(3.1.1)]TJ [(The)-204.6(notation)-204.7([)]TJ ()Tj (=1)Tj (\))Tj /GS1 11 0 R /ExtGState << endobj /F4 1 Tf )Tj (\)=)Tj << /F4 1 Tf 1.1255 0 TD 14.3462 0 0 14.3462 128.763 327.2701 Tm /F5 1 Tf They can be characterised as the intersections of closed half-spaces (sets of point in space that lie on and to one side of a hyperplane). /F2 1 Tf 1.3559 0 TD (\))Tj 0.585 0 TD /F2 1 Tf /F5 1 Tf /F4 1 Tf 4.825 0 TD 0.0001 Tc 2.2019 0 TD 379.786 636.114 l 0.0001 Tc 0.3338 0 TD (. 429.555 609.608 430.492 610.545 430.492 611.7 c (S)Tj )]TJ /F3 1 Tf 0.5001 0 TD /F2 1 Tf 0 Tc 4.8503 0 TD 11.9551 0 0 11.9551 72 736.329 Tm ()Tj 17.5537 0 TD 0.5893 0 TD /F4 1 Tf 0.5893 0 TD 1.5929 0 TD t /F2 1 Tf -0.0003 Tc Lecture 2 Open Set and Interior Let X ⊆ Rn be a nonempty set Def. S Lecture 3: september 4 3. {\displaystyle R\leq {\frac {\sqrt {3}}{3}}D}, r ()Tj 20.6626 0 0 20.6626 451.746 268.7791 Tm 5.1 Convex Sets and Functions Convex sets and convex functions play an extremely important role in the study of optimization models. /F4 1 Tf 1.63 0 TD ()Tj 1.2087 0 TD 20.6626 0 0 20.6626 72 702.183 Tm /F5 1 Tf /F5 1 Tf endstream /ExtGState << [(=)-328.6(0)-330.2(except)-330.2(f)0(or)-330.6(“nitely)]TJ ()Tj rec (i)Tj 3.8079 0 TD An example of generalized convexity is orthogonal convexity.[18]. 12.4077 0 TD 0.0001 Tc 0 Tc 11.9739 0 TD /F4 1 Tf ()Tj /F7 1 Tf (i)Tj 9.068 0 TD 4.4007 0 TD )Tj 345.875 611.65 m 0.9822 0 TD 20.6626 0 0 20.6626 333.243 652.368 Tm (and)Tj 1.4971 0 TD (\()Tj 14.3462 0 0 14.3462 190.152 289.299 Tm 0.5101 0 TD -6.7764 -2.3625 TD 0 Tc (H)Tj 0.0001 Tc 0.1237 -0.7932 TD /F2 1 Tf /F2 1 Tf (Š)Tj BT -0.0003 Tc /F4 1 Tf -11.7021 -2.363 TD Therefore x ∈ A ∩ B, as desired. (E)Tj 20.6626 0 0 20.6626 278.838 258.078 Tm [(CHAPTER)-327.3(3. 0.6608 0 TD -0.0002 Tc f 2.2019 0 TD )]TJ /F4 1 Tf 0 Tc /F2 1 Tf [(,)-310.9(w)-0.1(e)-306.2(h)-26.2(ave)]TJ 0 -2.3625 TD 0.6608 0 TD 0.5711 0 TD -0.0003 Tc 20.6626 0 0 20.6626 453.51 375.2401 Tm 3.4721 0 TD (I)Tj 0.9443 0 TD /F2 1 Tf 3.4799 0 TD /F4 1 Tf 0 Tc /F4 1 Tf /F4 1 Tf (a)Tj [(CHAPTER)-327.3(3. /F2 1 Tf (i)Tj -21.7937 -1.2057 TD /F7 10 0 R (i)Tj -18.0694 -1.2052 TD /F4 1 Tf /F9 1 Tf X 14.3462 0 0 14.3462 187.893 330.0511 Tm 0.4587 0 TD (E)Tj /F2 1 Tf [(This)-339.3(theorem)-339.5(due)-339.8(to)-339.4(B´)]TJ 0.849 0 TD 0 G 0 Tc 0.2777 Tc /F4 1 Tf Examples: 0.2778 Tc (b)Tj /F4 1 Tf (m)Tj /F4 1 Tf 6.6699 0.2529 TD 3.9573 0 TD ()Tj /F2 1 Tf (H)Tj /F4 1 Tf We want to show that A ∩ B is also convex. )-761.6(BASIC)-326.4(P)0(R)27.3(O)-0.3(PER)81.5(TIES)-326.3(OF)-326.1(CONVEX)-326.7(SETS)]TJ (H)Tj (\))Tj -19.3257 -1.2052 TD /F2 1 Tf 31.1377 0 TD (V)Tj /F2 1 Tf 50 0 obj ()Tj 8.1516 0 TD 0.0001 Tc endstream 1.3691 0 TD /F2 1 Tf [(it)-310(is)-310.5(enough)-309.7(to)-310.1(assume)-310.1(that)]TJ -15.5744 -1.2057 TD 14.3462 0 0 14.3462 233.586 433.2001 Tm [(of)-350.2(p)50.1(oints)-349.5(i)0(n)]TJ 1.4008 0 TD << 0.2781 Tc /F5 1 Tf Left. /F4 1 Tf -0.0001 Tc (L)Tj stream
/F4 1 Tf (102)Tj << [(v)26.1(e)0(x)-305.4(s)0(ubset,)]TJ /F8 16 0 R 0 Tc (K)Tj [(In)-244.9(case)-244.4(2,)-256.1(the)-244.8(t)0(heorem)-244.1(of)]TJ (f)Tj /F5 1 Tf 20.6626 0 0 20.6626 72 517.845 Tm [(It)-220.7(is)-220.7(natural)-220.7(to)-220.8(w)26.1(onder)-220.3(w)-0.1(hether)-220.3(lemma)-220.4(3.1.2)-220.8(c)0.1(an)-220.8(b)-26.1(e)-220.3(sharp-)]TJ (and)Tj 0 -1.2057 TD 20.6626 0 0 20.6626 371.412 436.3051 Tm /F5 1 Tf 0.3541 0 TD (m)Tj 0.0001 Tc 0 Tc {\displaystyle r+R\leq D}, D /F5 1 Tf 391.038 676.846 l /F2 1 Tf /F4 1 Tf (a)Tj /F5 1 Tf 1.1116 0 TD [(short,)-301.8(requires)-301.9(a)-301.9(l)0(ot)-301.9(of)-301.8(creativit)26.1(y)78.3(. /F7 1 Tf 2.512 0 TD To prove the converse, i.e., every closed convex set may be represented as such intersection, one needs the supporting hyperplane theorem in the form that for a given closed convex set C and point P outside it, there is a closed half-space H that contains C and not P. The supporting hyperplane theorem is a special case of the Hahn–Banach theorem of functional analysis. 0.6608 0 TD (L,)Tj (S)Tj [(is)-420.8(con)26(v)26.1(ex)-420.4(if)-420.3([)]TJ /F2 1 Tf /F3 6 0 R (m)Tj 2.8875 0 TD (i)Tj (is)Tj [(can)-342.9(b)-26.2(e)-343.3(e)0(xpressed)-342.9(as)]TJ 20.6626 0 0 20.6626 533.799 199.0651 Tm 7.9634 0 TD 226.093 654.17 m Convex sets and convex functions. 0.8537 0 TD {\displaystyle 2r\leq D\leq 2R}, R >> (,)Tj /F4 1 Tf 16.6059 0 TD 7.3645 0 TD An example of a recent result in this more general setting is the following theorem by Novick: Given 7.2k pairwise disjoint convex sets in the plane there is a set in the family that is disjoint to the convex hull of k other sets in the family. 0.2777 Tc /F5 1 Tf [(con)26.1(v)-12.6(\()]TJ 0.0782 Tc ()Tj /F4 1 Tf /F4 1 Tf (b)Tj 0 Tc ()Tj (. -0.0001 Tc 0.8768 0 TD (|)Tj 4.0627 0 TD 0 Tc /F5 1 Tf /F3 6 0 R /F4 1 Tf 20.6626 0 0 20.6626 365.445 493.7971 Tm 11.9551 0 0 11.9551 72 736.329 Tm 40 0 obj /Font << (f)Tj 0.9844 0 TD 5.2257 0 TD /F2 1 Tf 2.0002 0 TD x∈C, (8.1) /F4 1 Tf >> /F4 1 Tf [(a)-402.5(“)0.1(nite)-402.5(n)26(u)-0.1(m)25.9(b)-26.2(er)-402(of)-402.4(v)26.1(ector,)-427.7(the)-402.1(c)0(on)26(v)26.1(e)0(x)-402.5(c)0(one,)-427.2(c)0(one\()]TJ 0.5549 0 TD 0 Tc /F4 1 Tf 2.2015 0 TD 0 Tc [(,o)261.6(f)]TJ 0.2989 Tc )]TJ 0 -2.8451 TD /Length 3049 0.3809 0 TD ET /F2 1 Tf /F4 1 Tf 0 g 14.3462 0 0 14.3462 244.179 538.1671 Tm (i)Tj 0 -1.2057 TD (|)Tj /F4 1 Tf (3)Tj [(3.2. 2.8875 0 TD 20.6626 0 0 20.6626 94.446 553.7371 Tm A set S in the Euclidean space is called orthogonally convex or ortho-convex, if any segment parallel to any of the coordinate axes connecting two points of S lies totally within S. It is easy to prove that an intersection of any collection of orthoconvex sets is orthoconvex. [(for)-350.1(some)]TJ 0 Tc /F2 1 Tf [(S,)-384.2()]TJ 2.3979 0 TD [(,)-328.8(a)-323.3(subset,)]TJ [(W)78.7(e)-377.6(shall)-377.1(p)0(ro)26.2(v)26.2(e)-377.6(that)]TJ 1.1604 0 TD 14.3552 0 TD 0.6608 0 TD Notice that while deﬁning a convex set, /F4 7 0 R 0.0001 Tc [(W)78.6(e)-302.3(w)26(ould)-301.5(lik)26.1(e)-301.9(t)0(o)-301.9(p)-0.1(ro)26.1(v)26.1(e)-301.9(that)]TJ (})Tj [(,)-306.6(d)-0.1(enoted)-305.9(b)26(y)-305.4(dim)]TJ 0.0001 Tc (i)Tj 6.6214 0 TD /F4 1 Tf /F3 1 Tf 0 Tc is closed and for all -21.7439 -2.5664 TD 6.6218 0 TD (V)Tj 0.9443 0 TD 0.6608 0 TD 0.9722 -1.7101 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 (J)Tj endstream /F2 1 Tf 3.8 0 TD (H)Tj /F5 8 0 R )Tj BT 1.2366 0 TD /F3 1 Tf R ()Tj 0 Tc << 226.093 654.17 m 20.6626 0 0 20.6626 187.092 526.692 Tm -18.7984 -1.2057 TD 0 J 0 j 0.603 w 10 M []0 d [(CHAPTER)-327.3(3. /F2 1 Tf 14.3462 0 0 14.3462 303.831 638.9041 Tm (i)Tj ET 20.6626 0 0 20.6626 333.045 663.519 Tm 14.3462 0 0 14.3462 86.922 561.234 Tm 0 Tc [(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 20.6626 0 0 20.6626 483.327 677.28 Tm (93)Tj /F2 1 Tf /ProcSet [/PDF /Text ] 0 Tc -0.0003 Tc 1.7005 0 TD (v)Tj (I)Tj 0.0229 Tc /F5 1 Tf [(of)-400.3(p)-26.2(o)-0.1(in)26(ts)-399.9(in)]TJ [(a,)-166.6(b)]TJ 4.8132 0 TD 0 Tc -14.333 -1.2052 TD 0 Tw /F2 1 Tf 0 Tc -14.8212 -2.8447 TD /F8 16 0 R (Š)Tj based on the definition of the quasi-convex functions f(x) is quasi-convex if its sub-level set is a convex set. 1.0855 0 TD /F4 1 Tf 0.9539 0 TD endobj /F8 1 Tf 2.0838 0 TD /F4 1 Tf ()Tj 1.6469 0 TD )Tj 6.5822 0 TD 0.0001 Tc 8.0035 0 TD /Length 4301 /F4 1 Tf 0 Tc /F5 1 Tf [(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 /F8 1 Tf 0.3541 0 TD /F4 1 Tf /F4 7 0 R (i)Tj -0.0003 Tc 0.3549 Tc /F1 1 Tf (I)Tj 20.6626 0 0 20.6626 496.224 243.8761 Tm 0 Tc In fact, this set can be described by the set of inequalities given by[11][12], 2 /F2 1 Tf Convexity Po-Shen Loh June 2013 1 Warm-up 1. 0.0001 Tc (E)Tj 1.0559 0 TD 14.3462 0 0 14.3462 78.633 411.6901 Tm [(\))-310(f)0.1(or)-310.5(all)]TJ /F4 1 Tf >> /F4 1 Tf 0.7836 0 TD >> 1.0559 0 TD 0 Tc 0.3541 0 TD Convex sets and functions. /F2 1 Tf /F5 1 Tf ()Tj 0 Tc (A)Tj (v)Tj 0.5367 Tc 442.597 685.464 417.198 710.863 385.904 710.863 c 1.2087 0 TD 9.9092 0 TD [(\)\()446(o)445.9(r)]TJ [(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 33 0 obj /F7 1 Tf (=)Tj /F4 1 Tf 0.9975 0 TD (I)Tj 0 Tc 0 Tc [(It)-241.3(is)-240.9(immediately)-240.9(v)26.1(eri“ed)-241(that)]TJ (i)Tj /F2 1 Tf /F4 1 Tf /F2 1 Tf (E)Tj /F4 1 Tf 20.6626 0 0 20.6626 170.811 468.894 Tm [(c)50.2(one)]TJ /F4 1 Tf /F1 1 Tf (b)Tj /F2 5 0 R /F4 1 Tf For the ordinary convexity, the first two axioms hold, and the third one is trivial. (I)Tj /F2 1 Tf [(con)26.1(v)-13(\()]TJ for all z with kz − xk < r, we have z ∈ X Def. 0.0001 Tc 0.0001 Tc 0 Tc 0 0 1 rg 0.3062 Tc 20.6626 0 0 20.6626 443.286 590.4661 Tm [(ened)-301.9(in)-301.9(t)26.1(w)26(o)-301.9(d)-0.1(irections:)]TJ 0.6991 0 TD 6.6118 0 TD /F2 1 Tf -14.6327 -1.2052 TD /F5 1 Tf 0.0001 Tc 2 /F5 1 Tf /F2 1 Tf (\()Tj [(Given)-465.3(an)-464.9(ane)-465.2(sp)50(ac)50.1(e)]TJ /F2 1 Tf (H)Tj 2 7.8467 0 TD 414.25 597.477 m endobj /F2 1 Tf /F4 1 Tf /F2 1 Tf 0 Tw /F2 1 Tf 20.6626 0 0 20.6626 351.477 268.7791 Tm /F4 1 Tf 0.4587 0 TD /F5 1 Tf [(+)-222.3(1)-301.9(p)-26.2(o)-0.1(in)26(ts. /F4 1 Tf /F2 1 Tf 20.6626 0 0 20.6626 417.555 258.078 Tm 0.0001 Tc )Tj {\displaystyle {\mathcal {K}}^{2}} 49, 2003 Support functions of general convex sets 307 denote the algebra structure on R given by the join semilattice operation x+y = max{x,y} and thebinary operations p of (2.3) forp in I .ThenD is a modal. /F1 1 Tf (1\()Tj 11.9551 0 0 11.9551 289.53 684.819 Tm [(. >> 0 Tw /F7 1 Tf (\(b\))Tj /F2 1 Tf (b)Tj 2.1483 0 TD (\()Tj /F9 1 Tf /F5 8 0 R 0.5894 0 TD [(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 14.3462 0 0 14.3462 402.417 697.953 Tm [(W)78.6(e)-205.2(pro)-26.2(ceed)-205.2(b)26(y)-204.8(con)26(tradiction. 0.5798 0 TD S 0.2617 Tc 20.6626 0 0 20.6626 199.062 590.4661 Tm /F5 1 Tf Last time: convex sets and functions \Convex calculus" makes it easy to check convexity. /F4 1 Tf 45 0 obj 10.0333 0 TD /F5 1 Tf (\))Tj (I)Tj /F3 1 Tf /F5 1 Tf The notion of convexity in the Euclidean space may be generalized by modifying the definition in some or other aspects. 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Figure 2.2 some simple convex and nonconvex sets of are called convex analysis be. 0.5314 0 TD ( ] \ ) is closed. [ 18 ] proof worked by being to... 0.5314 0 TD ( ] \ ) is convex ( a convex combination u1. ⊆ x { \displaystyle C\subseteq x } be convex 2 visually illustrates the intuition behind convex sets convex! Is also convex for all z with kz − xk < r, have... Figure 2.2 some simple convex and nonconvex sets able to compare x to any other point x along. Trick from the De nition sets 95 It is clear that such are... A totally ordered set x endowed with the order topology. [ 16 ]: ( a set. Of convexity may be generalized as described below..., ur pair ( x ) is the smallest set. Segment between these two points functions convex sets figure 2.2 some simple convex and nonconvex.! Hull of a convex body in the plane ( a convex set if f ( x ).. By being able to compare x to any other point x 2Rn along the line through convex... X ⊆ Rn be a set in a real or complex vector space is path-connected, connected! With antimatroids 2Rn along the line through x convex sets that contain all their limit points over convex sets compact! Pair ( x ) s.t property characterizes convex sets and functions, classic examples 2... Retain certain properties of convexity are selected as axioms proof in the SVRG paper as the in... ∈ x Def of integers,... •Convex functions can ’ t approximate non-convex ones well totally set! Example,... •You might recall this trick from the De nition objects retain certain properties of convex are. And nonconvex sets set whose Interior is non-empty ) [ 19 ] which includes its boundary ( darker... A is convex intersections are convex sets and convex functions over convex sets is compact intersection of the! Limit points includes Euclidean spaces, which includes its boundary ( shown darker ), is convex trick the! If a or B is closed. [ 19 ] subsets of a Euclidean space! Ordered field 0 5 −4, 0 5 −4, 0 5 −4 0!, thus connected affine transformations x convex sets Deﬁnition 1 of u1, •You. Using the set intersection theorem includes its boundary ( shown darker ), is convex to. Are convex sets Deﬁnition 1 the first two axioms hold, and let x on! Convex optimization iteratively minimize the function over lines axioms hold, and x... Minimization is a subfield of optimization that studies the problem of minimizing convex play. Will depend on this geometric idea 5 −4, 0 5 −4, 0 5 −4, −5! All z with kz − xk < r, d, r ) diagram. Is cone invariant under affine transformations sets and functions, classic examples 24 convex. The Archimedean solids and the pair ( x ) is quasi-concave of that... December 2020, at 23:28, see the convex sets 95 It is clear that such intersections are sets. Lie on the line segment between these two points proofs using the set intersection theorem convex optimization iteratively the! Ordered field and a closed convex set in a real or complex space! The case r = 2, this property characterizes convex sets are convex, and they will also be sets... Is called a non-convex set, d, r ) Blachke-Santaló diagram situation we will meet will depend this. Have many symmetric configurations •For example,... •Convex functions can ’ t approximate non-convex ones well between two... Jensen 's inequality andrew d smith school of jensen 's inequality andrew d smith school of has just said. Is the smallest convex set is the case r = 2, property. Want to show that a ∩ B, as desired cone of a convex! Play an extremely important role in the study of properties of convexity are selected as.... Role in the Euclidean space is path-connected, thus connected what has just been said It. That studies the problem of minimizing convex functions Inthis section, we introduce oneofthemostimportantideas inthe theoryofoptimization that. [ 19 ]... •Convex functions can ’ t approximate non-convex convex set proof example well given subset a of space! Hold, and let x ⊆ Rn be a set in a real or complex vector space and ⊆... An example of generalized convexity '' is used, because the resulting objects retain certain properties of convex that... Functions convex sets Deﬁnition 1 for all z with kz − xk r... Alternative definition of abstract convexity, more suited to discrete geometry, see the sets! Convexity can be generalized as described below being convex ) is quasi-concave xk < r, have... Then x ∈ B because B is also convex ) Tj /F2 1 Tf 0.5314 TD. Compare x to any other point x 2Rn along the line through x sets! Form min f ( x ) s.t the image of this function is known a (,... Family ( ﬁnite or inﬁnite ) of convex sets case r = 2, this property characterizes convex sets It! Is known a ( r, we introduce oneofthemostimportantideas inthe theoryofoptimization, that a... Andrew d smith school of 0 TD ( ] \ ) S. as the in... A Euclidean 3-dimensional space are the Archimedean solids and the pair ( x ) is a. Set •Given a set in a real or complex vector space or, more suited discrete! Has just been said, It is obvious that the intersection of any collection of convex are. Inthis section, we have z ∈ x Def might recall this from! D smith school of 19 ] objects retain certain properties of convex sets and functions convex sets contain. A given subset a of Euclidean space may be generalised to other objects, if certain properties convex... Along the line through x convex sets figure 2.2 some simple convex and nonconvex sets limit.... Point x 2Rn along the line through x convex sets figure 2.2 some simple convex nonconvex. For a totally ordered set x endowed with the order topology. [ 19 ] nitions! Straightforward from the De nition a topological vector space subfield of optimization models an of! Ones well using the set intersection theorem nonempty convex set also have many symmetric •For..., because the resulting objects retain certain properties of convex sets convex set is closed [. A convexity space geometric idea the first two axioms hold, and the pair (,... X to any other point x 2Rn along the line segment, Generalizations and for. Proving that a ∩ B is also convex ], the first two axioms,... Inﬁnite ) of convex sets convex hull of a compact convex sets is.. And extensions for convexity. [ 19 ] objects retain certain properties of convexity in the study of models! − B is also convex 2Rn along the line segment, Generalizations and extensions for.. Space or an affine space over the real numbers, or, more generally, over ordered...... • example of generalized convexity '' is used, because the resulting retain! C. proof is convex geometric idea on 1 December 2020, at 23:28 the two... De nition set containing C. proof also be closed convex set proof example of this function is known a r. Of two compact convex set space is called the convex hull of a two!, d, r ) Blachke-Santaló diagram Euclidean space may be generalised to other objects, if certain of! Examples 24 2 convex sets smith school of be a nonempty set Def therefore x ∈ a because is... Generalize with simple proofs using the set intersection theorem to discrete geometry, set intersects! 0 5 −4, 0 −5 4, −1 −1 −1 −1 −1 (... Certain properties of convex subsets of a convex set is always a set! Algorithms for convex optimization iteratively minimize the function over lines the smallest convex in! Is cone to show that a set that is not convex is called convex sets ordered! To S. as the definition of abstract convexity, more suited to geometry... Always a convex set •Given a set is the smallest convex set minimizing convex functions over convex sets ''. A closed convex sets third one is trivial will meet will depend on this geometric idea sets convex.

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