0.889 0 TD /F2 1 Tf 0.5314 0 TD /F4 1 Tf 1.5929 0 TD 0.5763 0 TD 0 0 1 rg ()Tj 0.8163 0 TD /F2 1 Tf 0.0001 Tc 329.211 654.17 l /F2 1 Tf >> The notion of a convex set can be generalized as described below. /F4 1 Tf 0 -1.2052 TD [(])-301.7(o)0(r)-301.8(L)0.2(ang)-301.8([)]TJ 3.4721 0 TD /F1 1 Tf 0 J 0 j 1.494 w 10 M []0 d 20.6626 0 0 20.6626 316.746 258.078 Tm D (S)Tj 0.3541 0 TD /F4 1 Tf 0.49 0 TD ()Tj 0 Tc (\()Tj >> /F7 10 0 R -21.7937 -1.2057 TD 0 Tc [(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 For the ordinary convexity, the first two axioms hold, and the third one is trivial. 20.6626 0 0 20.6626 333.243 652.368 Tm 0 Tc /F2 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 1.0903 0 TD 0.0588 Tc (i)Tj [(,)-315.4(t)0.2(hat)-306.9(is,)]TJ /F4 1 Tf /F3 1 Tf /GS1 11 0 R 9.1665 0 TD 0 -1.2052 TD endstream 1.0955 0 TD 0 Tc /F2 1 Tf 414.25 625.823 m 0.849 0 TD 0.0001 Tc [(con)26.1(v)-13(\()]TJ (If)Tj 2.1483 0 TD (i)Tj [(Car)50.1(a)-0.1(th)24.8(´)]TJ ()Tj ()Tj 0 -1.2057 TD 1.2087 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. /F2 1 Tf 1.8064 0 TD (C)Tj In a real vector-space, the Minkowski sum of two (non-empty) sets, S1 and S2, is defined to be the set S1 + S2 formed by the addition of vectors element-wise from the summand-sets, More generally, the Minkowski sum of a finite family of (non-empty) sets Sn is the set formed by element-wise addition of vectors, For Minkowski addition, the zero set {0} containing only the zero vector 0 has special importance: For every non-empty subset S of a vector space, in algebraic terminology, {0} is the identity element of Minkowski addition (on the collection of non-empty sets).[13]. 0.3337 0 TD 1.9453 0 TD (H)Tj (\()Tj 329.211 654.17 l 1.6025 0 TD /F3 1 Tf /F3 1 Tf 0 -1.2057 TD 20.6626 0 0 20.6626 351.477 268.7791 Tm [(,t)377.6(h)377.5(e)]TJ S 14.3462 0 0 14.3462 343.818 380.181 Tm 0.3541 0 TD 0.0001 Tc 0 0 1 rg ()Tj /F5 1 Tf (i)Tj )Tj (2)Tj (V)Tj /F3 1 Tf 0 Tc /F2 1 Tf (\)=)Tj 0.3338 0 TD )]TJ /F2 1 Tf >> ET (i)Tj 0 g 40 0 obj 1 i 0 Tc ()Tj rec 5.5036 0 TD [(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 0 Tc -13.3009 -3.3269 TD 0.1667 Tc 14.3462 0 0 14.3462 389.178 649.272 Tm /F2 5 0 R 20.6626 0 0 20.6626 404.523 652.368 Tm (A)Tj [(3.2)-1125.1(C)0.1(arath)24.3(«)]TJ (i)Tj 4.0253 0 TD (subsets)Tj /F4 1 Tf 20.6626 0 0 20.6626 421.299 541.272 Tm (q)Tj (i)Tj /F4 1 Tf /F4 1 Tf 1.0554 0 TD (\))Tj [(\)\()446(o)445.9(r)]TJ 0.7919 0 TD (v)Tj /F8 1 Tf /F6 9 0 R /F2 1 Tf 0.5001 0 TD An example of generalized convexity is orthogonal convexity.[18]. 0.6943 0 TD 0.2778 Tc 4.7126 0 TD 6.6161 0 TD (b)Tj ET [(\(1\))-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 b 0.0001 Tc /F2 1 Tf (i)Tj (a)Tj /F4 1 Tf /F2 1 Tf /GS1 gs 0 Tc (i)Tj (|)Tj 0.3541 0 TD (f)Tj (S)Tj 0.9443 0 TD 14.3462 0 0 14.3462 460.827 372.144 Tm [(,)-448.7(for)]TJ ()Tj 14.3462 0 0 14.3462 119.646 433.2001 Tm )Tj 0 Tc [(=)-328.6(0)-330.2(except)-330.2(f)0(or)-330.6(“nitely)]TJ (i)Tj /F3 1 Tf 20.6626 0 0 20.6626 513.189 701.0491 Tm 0.9052 0 TD 0 Tc 0.6669 0 TD 1.2216 0.7187 TD (,)Tj Equivalently, a convex set or a convex region is a subset that intersect every line into a single line segment (possibly empty). 0 Tc (f)Tj 0.5101 0 TD [(CHAPTER)-327.3(3. 0 Tc /F2 1 Tf (b)Tj >> [(\). 4.7701 0 TD /F2 1 Tf /F3 1 Tf 0.8564 0 TD (95)Tj 9.9253 0 TD /F4 1 Tf (\()Tj /F3 1 Tf (E)Tj /F4 1 Tf ()Tj 0.788 0 TD 20.6626 0 0 20.6626 249.741 576.498 Tm [(called)-301.9(a)]TJ 0.5711 0 TD 0.0001 Tc 0.5798 0 TD endobj 4.0627 0 TD 0 Tc (+1)Tj [(1o)393.7(ft)393.8(h)393.7(e)]TJ 0.3615 Tc 0 Tw + (A)Tj X -5.1077 -1.7841 TD /F3 1 Tf /F1 1 Tf for all z with kz − xk < r, we have z ∈ X Def. /F2 1 Tf /F4 1 Tf /F4 1 Tf 0.585 0 TD The subspace Y is a convex set if for each pair of points a, b in Y such that a ≤ b, the interval [a, b] = {x ∈ X | a ≤ x ≤ b} is contained in Y. 1.4958 0 TD (93)Tj 2.3164 0 TD Vol. -0.0001 Tc /F2 1 Tf -14.333 -1.2052 TD (+)Tj -7.6907 -2.3625 TD T* /F4 1 Tf ET /F2 1 Tf [(eo)50.1(dory)-350.3(t)0.2(he)50.2(or)50.2(em)]TJ {\displaystyle R\leq {\frac {\sqrt {3}}{3}}D}, r [(W)78.7(e)-377.6(shall)-377.1(p)0(ro)26.2(v)26.2(e)-377.6(that)]TJ /F3 1 Tf /F2 1 Tf Math 484: Nonlinear Programming1 Mikhail Lavrov Chapter 2, Lecture 1: Convex sets February 4, 2019 University of Illinois at Urbana-Champaign 1 Convexity Earlier this semester, we showed that if x is a critical point of f: Rn!R, and Hf(x) 0 for all x 2Rn, then x is a global minimizer. Some examples of convex subsets of a Euclidean 3-dimensional space are the Archimedean solids and the Platonic solids. /F2 1 Tf (i)Tj /F7 10 0 R 20.6626 0 0 20.6626 208.116 378.372 Tm /F4 1 Tf [(is)-301.9(allo)26.1(w)26(e)0(d\). /F4 1 Tf >> 0.9857 0 TD [(a,)-166.6(b)]TJ 0.0001 Tc /F5 1 Tf 0.5314 0 TD (Š)Tj (S)Tj 226.093 597.477 l (\()Tj >> 220.959 620.154 l 0 Tc /F2 1 Tf /F3 1 Tf 0.0001 Tc ()Tj 0.446 Tc /F2 1 Tf << >> The following famous theorem, proved by Dieudonné in 1966, gives a sufficient condition for the difference of two closed convex subsets to be closed. 21.7941 0 TD /F3 1 Tf ()Tj /F2 1 Tf (de“ning)Tj 0.0001 Tc 0 g A set C Rnis convex if 8x1;x2 2C;8 2[0;1] we have that x = x1 +(1 )x2 2C: Intuitively, a set is convex if the line segment between any two of its points is in the set. 6.6699 0.2529 TD 0 Tc 0 Tc /F4 1 Tf [(The)-437.3(answ)26(e)0(r)-436.9(i)0(s)-437.8(y)26.1(es)-437.3(in)-437.4(b)-26.2(o)-0.1(th)-437(cases. /F4 1 Tf /F3 1 Tf 0.3338 0 TD (V)Tj 20.6626 0 0 20.6626 140.004 436.3051 Tm >> /F4 1 Tf /F2 1 Tf 0.0001 Tc 329.211 597.477 m 4.8001 0 TD 45 0 obj 0.8563 0 TD (i)Tj /F4 1 Tf -0.0001 Tc /F2 1 Tf /F2 1 Tf /F4 1 Tf [(man)26.2(y)]TJ -0.0001 Tc (I)Tj 0.585 0 TD ≤ (i)Tj /F1 1 Tf ()Tj 0 0 1 rg 0 Tc 0.5001 0 TD 7.053 0 TD (i)Tj 2.8875 0 TD /F4 1 Tf 14.7128 0 TD 0 Tw (m)Tj (E)Tj 1.3691 0 TD /F4 1 Tf (R)Tj /F2 1 Tf [(only)-376.7(d)0(ep)-26.1(ends)-376.2(on)-376.8(the)]TJ 0.7814 0 TD [(Con)26(v)26.1(ex)-424.8(sets)-425.1(also)-424.7(arise)-425.1(in)-425.2(terms)-424.7(o)-0.1(f)-425.1(h)26(yp)-26.2(erplanes. 0.7836 0 TD [(a,)-166.6(b)]TJ BT [(is)-370.9(anely)-371.2(dep)50.1(e)0.1(ndent)-371(i)-371.2(ther)50.2(e)-371(i)0(s)-371.3(a)-371.1(family)]TJ (\))Tj /F4 1 Tf stream 0.5893 0 TD 20.6626 0 0 20.6626 355.869 663.519 Tm 0.2777 Tc /F7 10 0 R (,)Tj 1.0789 0 TD 0.2775 Tc /F3 1 Tf [(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 0 Tc 14.3462 0 0 14.3462 109.458 587.3701 Tm (I)Tj -17.8918 -2.4431 TD /F2 1 Tf /F5 1 Tf /F7 1 Tf 0 Tc 0.2777 Tc /F2 1 Tf )-762.5(CONVEX)-326(SETS)]TJ The notion of convexity in the Euclidean space may be generalized by modifying the definition in some or other aspects. )-762.6(CARA)81.1(TH)]TJ 414.25 597.477 m 14.3462 0 0 14.3462 490.644 674.175 Tm 1.0559 0 TD B [(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 (i)Tj 2.1 Convex hull Convex hull of a set of points C(denoted Conv(C)) is all possible convex combinations of the subsets of C. It is clear that the convex hull is a convex set. 0.2223 Tc 0.9443 0 TD Every subset A of the vector space is contained within a smallest convex set (called the convex hull of A), namely the intersection of all convex sets containing A. (f)Tj [(Lemma)-375.4(3.2.1)]TJ 0 0 1 rg (b)Tj (S)Tj 0 g 0 -1.2057 TD [(tices)-301.9(b)-26.2(elong)-301.9(to)]TJ (S)Tj (Š)Tj /F2 1 Tf (L)Tj /F7 1 Tf 0 Tc Minkowski addition behaves well with respect to the operation of taking convex hulls, as shown by the following proposition: Let S1, S2 be subsets of a real vector-space, the convex hull of their Minkowski sum is the Minkowski sum of their convex hulls. /F5 1 Tf /F3 1 Tf /ProcSet [/PDF /Text ] /F4 1 Tf /F2 1 Tf /F7 1 Tf /F8 1 Tf 14.3462 0 0 14.3462 196.695 403.1671 Tm /F7 1 Tf [(CHAPTER)-327.3(3. 0.2781 Tc 1.0528 0 TD /F4 7 0 R (If)Tj (1)Tj 0 Tc 0.6904 0 TD -0.0003 Tc (. 3.2007 0 TD )Tj (. /F4 1 Tf 0.8341 0 TD 2.0442 0 TD (\()Tj ()Tj [(that)-224.8(the)-224.4(a)-0.1(ne)-224.8(s)0(pace)]TJ BT /F5 1 Tf 0.3541 0 TD /Length 6066 /F7 1 Tf [(eo)50.1(dory�s)]TJ 11.9551 0 0 11.9551 72 736.329 Tm /F4 1 Tf Figure 3.1: Example of a convex set (left) and a non-convex set (right). /F4 1 Tf Chapter 8 Convex Optimization 8.1 Deﬁnition Aconvexoptimization problem (or just a convexproblem) is a problem consisting of min- imizing a convex function over a convex set. (+)Tj 7.8467 0 TD 442.597 597.477 l /F5 1 Tf (\()Tj (H)Tj )Tj /F3 1 Tf (E)Tj /F2 1 Tf 1.494 w /F2 1 Tf 3.8 0 TD 1.1534 0 TD /F2 1 Tf ()Tj 20.6626 0 0 20.6626 120.879 590.4661 Tm 0 Tc /F7 1 Tf /F2 1 Tf 1.2366 0 TD ()Tj /F5 1 Tf A convex set is not connected in general: a counter-example is given by the space Q, which is both convex and totally disconnected. [(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 -21.1681 -1.2057 TD /F3 1 Tf 0 1 0 rg 21.1364 0 TD (i)Tj /F2 1 Tf /F2 1 Tf /F4 1 Tf ) 5.9074 0 TD /F4 1 Tf 20.6626 0 0 20.6626 182.34 541.272 Tm C 8.1141 0 TD (f)Tj << D (V)Tj 1.2549 0 TD -6.7764 -2.3625 TD /F4 1 Tf 379.485 636.416 m (H)Tj /F2 1 Tf 20.6626 0 0 20.6626 163.062 576.498 Tm 1.6295 0 TD D /F4 1 Tf (i)Tj Closed convex sets are convex sets that contain all their limit points. (=0)Tj ()Tj 387.355 629.139 m /F5 1 Tf /F2 1 Tf /ProcSet [/PDF /Text ] 1.4971 0 TD 0.6608 0 TD 0.2775 Tc 2.644 0 TD [(CHAPTER)-327.3(3. /F4 1 Tf 1.6021 0 TD /F7 1 Tf /F2 1 Tf (,...,m)Tj /F4 7 0 R 20.6626 0 0 20.6626 195.444 292.4041 Tm 0 Tc 20.6626 0 0 20.6626 278.838 258.078 Tm (sets,)Tj 0 Tc [(In)-244.9(case)-244.4(2,)-256.1(the)-244.8(t)0(heorem)-244.1(of)]TJ /F2 1 Tf 0.0001 Tc 14.3462 0 0 14.3462 311.571 191.9641 Tm /F5 1 Tf 226.093 654.17 m 0.7366 0 TD (f)Tj 0 Tc )-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 is a linear subspace. (H)Tj /F4 1 Tf 1.6291 0 TD -0.1302 -0.2529 TD Concretely the solution set to (4.6) is cone. 20.6626 0 0 20.6626 72 701.0491 Tm (m)Tj /F2 1 Tf ([)Tj /F2 1 Tf (K)Tj /F4 1 Tf It is the smallest convex set containing A. (ing)Tj Convex set Deﬁnition A set C is called convexif x,y∈ C =⇒ x+(1 − )y∈ C ∀ ∈ [0,1] In other words, a set C is convex if the line segment between any two points in C lies in C. Convex set: examples Figure: Examples of convex and nonconvex sets. /Length 5100 (E,)Tj [(asserts)-244.4(that)]TJ 20.6626 0 0 20.6626 355.896 383.277 Tm /F4 1 Tf /F4 1 Tf ()Tj (of)Tj 0 Tc 0.8537 0 TD 7.3645 0 TD -17.8646 -1.2052 TD [(EODOR)81.5(Y)0(’S)-326.3(THEOREM)]TJ endstream The common name "generalized convexity" is used, because the resulting objects retain certain properties of convex sets. 0 Tc (,)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 /F10 24 0 R [(is)-353.6(an)26.1(y)-353.7(nonconstan)26.1(t)-353.6(ane)]TJ /F4 1 Tf /F1 1 Tf S 1.1691 0 TD (\))Tj /F9 1 Tf ()Tj 112.707 597.477 m /F4 1 Tf 0.0001 Tc /ExtGState << (. b 14.3462 0 0 14.3462 181.8 523.587 Tm 1.9361 0 TD •Neural nets also have many symmetric configurations •For example, ... •You might recall this trick from the proof in the SVRG paper. (0)Tj /Font << [(line)50.2(ar)-365.8(c)50.2(o)0(mbinations)]TJ /F4 1 Tf [1][2] /F4 1 Tf 0.0001 Tc 0.0001 Tc (,...,a)Tj 0.9443 0 TD 0 Tc /F4 1 Tf ()Tj /F9 1 Tf 0 Tc -22.3496 -1.2052 TD 0.0001 Tc /F4 1 Tf ()Tj /ExtGState << 14.3462 0 0 14.3462 325.017 573.402 Tm /F4 1 Tf /F5 1 Tf 3.6454 0 TD 13.4618 0 TD /F8 16 0 R >> ()Tj 0.5893 0 TD /F2 1 Tf 20.6626 0 0 20.6626 72 702.183 Tm -18.5359 -1.2052 TD = /F1 1 Tf [(c)50.2(o)0(mbinations)-349.5(of)-349.8(families)-349.5(o)0(f)]TJ /F3 1 Tf /F4 1 Tf /F2 1 Tf 0.389 0 TD 0.3541 0 TD /F7 1 Tf (\). 379.485 628.847 m /F3 1 Tf 0 Tc [(hul)-50.1(l)]TJ 7.9701 0 0 7.9701 299.232 612.162 Tm 14.3462 0 0 14.3462 340.056 265.683 Tm 2.0002 0 TD 0.0001 Tc /F4 1 Tf (|)Tj 0.6608 0 TD /F4 1 Tf (R)Tj 0.0001 Tc /F4 1 Tf (E)Tj (H)Tj [(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. 0 g 1.8059 0 TD /F1 1 Tf /F2 1 Tf /F2 1 Tf /F4 1 Tf (Š)Tj (,)Tj (I)Tj /Font << 0.5763 0 TD (|)Tj -14.8212 -2.8447 TD 0.0001 Tc [(EODOR)81.5(Y)0(’S)-326.3(THEOREM)]TJ /F3 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 ()Tj [(of)-400.3(p)-26.2(o)-0.1(in)26(ts)-399.9(in)]TJ (´)Tj (94)Tj /F2 1 Tf /F5 1 Tf 220.959 705.193 l 0 g 2.4384 0 TD /F4 1 Tf The convex subsets of R (the set of real numbers) are the intervals and the points of R. Some examples of convex subsets of the Euclidean plane are solid regular polygons, solid triangles, and intersections of solid triangles. /F5 1 Tf (i)Tj )]TJ 0.3541 0 TD 0 Tc [(,i)354.9(s)10.4(a)]TJ >> 0.3809 0 TD 14.3462 0 0 14.3462 102.546 540.5161 Tm 0 Tc 20.3985 0 TD -0.0002 Tc 0 Tc )-762.5(CONVEX)-326(SETS)]TJ 0.1666 Tc (X)Tj (i)Tj -16.2673 -1.2057 TD << /F1 1 Tf 0 Tc /F2 1 Tf 0.3541 0 TD /F3 1 Tf endobj stream 14.3552 0 TD /Length 5598 0 Tc [(con)26.1(t)0.1(aining)]TJ 0.5894 0 TD [(=K)277.5(e)277.7(r)]TJ 20.6626 0 0 20.6626 453.762 626.313 Tm )]TJ /F2 1 Tf − 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. (i)Tj /F4 1 Tf 0.389 0 TD 20.6626 0 0 20.6626 132.705 543.6121 Tm (H)Tj (i)Tj 0.5001 0 TD /F4 1 Tf /F1 1 Tf /F2 1 Tf /F2 1 Tf ()Tj )-762.6(CARA)81.1(TH)]TJ ()Tj [(ve)50.1(ctors)-306.9(i)-0.1(n)]TJ ( )Tj (1)Tj (S)Tj /F7 10 0 R /F4 1 Tf 0 -1.2057 TD (\()Tj /F4 1 Tf R 0.9152 0 TD >> 1.2209 0 TD /F5 8 0 R The elements of are called convex sets and the pair (X, ) is called a convexity space. /ProcSet [/PDF /Text ] (a)Tj /F7 1 Tf 20.6626 0 0 20.6626 221.58 541.272 Tm /F2 1 Tf The theorem simpli es many basic proofs in convex analysis but it does not usually make veri cation of convexity that much easier as the condition needs to hold for all lines (and we have in nitely many). 0.5798 0 TD /F7 1 Tf [(family)-342.4(of)-342.8(half-spaces)-342.4(asso)-26.2(ciated)-342.4(with)-342.4(h)26(y)-0.1(p)-26.2(erplanes)-342.4(p)-0.1(la)26.1(y)-342.4(a)]TJ -0.0001 Tc /F4 1 Tf [(i,)-166.5(j)]TJ /Font << ({)Tj (i)Tj 0 -1.2052 TD 0.0001 Tc /F3 1 Tf (=)Tj /F4 1 Tf 0.0001 Tc (and)Tj ET 442.597 654.17 m 20.6626 0 0 20.6626 94.833 242.5891 Tm /F2 1 Tf {\displaystyle D^{2}{\sqrt {4R^{2}-D^{2}}}\leq 2R(2R+{\sqrt {4R^{2}-D^{2}}})}, and can be visualized as the image of the function g that maps a convex body to the R2 point given by (r/R, D/2R). /F4 1 Tf [(,)-315.4(t)0.2(he)-306.5(c)50.2(one,)]TJ [(eo)50.1(dory�s)-350(T)0.1(he)50.2(or)50.2(em)]TJ ()Tj ()Tj /F5 1 Tf 0.0001 Tc 0.8564 0 TD (\))Tj 0.5893 0 TD 3 0 Tc 0 g (i)Tj 354.609 710.863 329.211 685.464 329.211 654.17 c (cone\()Tj 1.0554 0 TD (\()Tj [(hul)-50.1(l)]TJ Equivalently, a convex set or a convex region is a subset that intersect every line into a single line segment (possibly empty). 0.0001 Tc 0 g 0 Tc /F2 1 Tf 345.875 611.65 m 20.6626 0 0 20.6626 443.367 529.6981 Tm /F4 7 0 R 0.6773 0 TD 5.2234 -1.7841 TD >> 5.5102 0 TD (with)Tj ()Tj [(p)-26.2(o)-0.1(in)26(ts)]TJ 379.786 636.114 l (103)Tj 0.3541 0 TD 0 -1.2052 TD /F4 1 Tf (})Tj (´)Tj rec 0.0001 Tc 0 g (+1)Tj 0.5893 0 TD /F4 1 Tf 0.3541 0 TD /F5 1 Tf /F2 1 Tf [(\),)-423.8(but)-399(if)]TJ 0.6669 0 TD A vector x0 is an interior point of the set X, if there is a ball B(x0,r) contained entirely in the set X Def. << 0 Tc /F7 1 Tf /F4 1 Tf [(is)-306.4(e)50.2(q)0.1(ual)]TJ 0 Tc if f(x,y) is convex in (x,y) and C is a convex set, then g(x) = inf y∈C f(x,y) is convex examples • f(x,y) = xTAx+2xTBy +yTCy with A B BT C 0, C ≻ 0 minimizing over y gives g(x) = infy f(x,y) = xT(A−BC−1BT)x g is convex, hence Schur complement A−BC−1BT 0 • distance to a … /F2 1 Tf /F4 1 Tf 0.9073 0 TD /F7 1 Tf /F2 1 Tf 0 -1.2052 TD ()Tj 14.3462 0 0 14.3462 478.044 674.175 Tm /F11 25 0 R (|)Tj (Š)Tj Use the set intersection theorem, and existence of optimal solution <=> nonemptiness of \(nonempty level sets) Example 1: The set of minima of a closed convex function f over a closed set X is nonempty if there is no asymptotic direction of X that is a 0 Tc (A)Tj 0.5101 0 TD /F3 1 Tf 0.0041 Tc /GS1 11 0 R 0.1667 Tc (. 0.5101 0 TD 20.6626 0 0 20.6626 199.062 590.4661 Tm ()Tj /F4 1 Tf /F1 4 0 R 0.3541 0 TD 2 Simple examples of convex sets are: The empty set ;, the singleton set fx 0g, and the complete space Rn; Lines faT x= bg, line segments, hyperplanes fAT x= bg, and halfspaces fAT x bg; Euclidian balls B(x 0; ) = fxjjjx x 0jj 2 g. 0.9974 0.7501 TD 0 Tc 0 Tc (1\()Tj (v)Tj ()Tj /F4 1 Tf 0 -1.2052 TD /F7 1 Tf (S)Tj 3 0 obj 0.6991 0 TD %âãÏÓ )-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 /F5 1 Tf )]TJ /F3 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. 2.8204 0 TD (})Tj /GS1 gs 14.3462 0 0 14.3462 448.479 623.217 Tm [(p)50(oints,)]TJ /F2 1 Tf 3.3096 0 TD 0 Tc /F4 1 Tf /F2 1 Tf (S)Tj /F2 1 Tf [(CHAPTER)-327.3(3. 0.0001 Tc 0 Tc /F4 1 Tf 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 g ()Tj (S)Tj 0.3338 0 TD ({)Tj -0.0001 Tc /F4 1 Tf /F2 1 Tf ET (S)Tj (m)Tj 0.0002 Tc /ExtGState << (q)Tj ()Tj /F4 1 Tf (a)Tj 14.3462 0 0 14.3462 343.449 239.493 Tm R 0.5893 0 TD 0 Tw (\()Tj ()Tj 14.3462 0 0 14.3462 517.824 540.5161 Tm 0 Tc 0.5893 0 TD )-762.6(CARA)81.1(TH)]TJ /GS1 11 0 R /F2 1 Tf Some other properties of convex sets are valid as well. 1.525 0 TD Proof. /F3 1 Tf 0.6608 0 TD [(Bounded)-263.2(c)0(on)26(v)26.1(e)0(x)-263.2(sets)-263.5(arising)-263.6(a)-0.1(s)-263.1(t)0(he)-263.6(in)26(tersection)-263.2(o)-0.1(f)-263.5(a)-263.6(“nite)]TJ 0 Tc 0.9443 0 TD (I)Tj [(,s)315.1(p)365(a)314.9(n)314.8(n)314.9(e)365.1(d)8.3(b)315(y)]TJ ()Tj 2.5634 0 TD 4.8503 0 TD [(m)25.9(u)-0.1(st)-391.6(b)-26.2(e)-392(anely)]TJ 1.1116 0 TD (\)=)Tj 11.9551 0 0 11.9551 378.099 572.1901 Tm 2.3979 0 TD (i)Tj (Š)Tj (\(with)Tj /F8 1 Tf 0.3938 Tc /F7 1 Tf 6.5822 0 TD /F8 1 Tf /F4 1 Tf /F5 8 0 R /F7 1 Tf 0.75 g 0.0001 Tc 0.612 0 TD 6.0843 0 TD -18.0694 -1.2052 TD 0 -1.2052 TD /F2 1 Tf << /F5 1 Tf /F4 1 Tf /F2 1 Tf )-761.6(BASIC)-326.4(P)0(R)27.3(O)-0.3(PER)81.5(TIES)-326.3(OF)-326.1(CONVEX)-326.7(SETS)]TJ 14.3462 0 0 14.3462 377.244 490.701 Tm -19.1628 -1.2057 TD /F3 6 0 R (\))Tj Lecture 3: september 4 3. [(p)-26.2(o)-0.1(in)26(ts)-301.9(in)]TJ /F2 1 Tf 30 0 obj [(of)-388(a)-388.1(nonempt)26.2(y)-387.6(con-)]TJ 20.6626 0 0 20.6626 365.445 493.7971 Tm (. /F4 1 Tf )-775.3(Giv)26.1(e)0(n)]TJ ()Tj stream (i)Tj /F4 1 Tf 0.0001 Tc ⊆ 9.9092 0 TD /F2 1 Tf 0.2781 Tc 11.9551 0 0 11.9551 72 736.329 Tm /F7 1 Tf 0.2226 Tc If the feasible region is a convex set, and if the objective function is a convex function, ... detailed proofs of these statements but in my opinion they are not particularly instructive given 0 -2.3625 TD /F2 5 0 R 0 Tw [(,)-487.5(t)0.1(hen)-460.5(t)0.1(her)50.1(e)-460.8(exists)-460.3(a)-460.5(s)0.1(e)50.1(q)0(uenc)50.1(e)-460.8(o)-0.1(f)]TJ (J)Tj /ExtGState << 0.5549 0 TD 0.389 0 TD 0.9622 0 TD 20.6626 0 0 20.6626 255.204 663.519 Tm [(\(close)50.1(d\))-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 13.4618 0 TD 0.849 0 TD 357.557 625.823 l /F2 1 Tf (m)Tj 0.6608 0 TD (. /F7 10 0 R )Tj /F4 1 Tf (b)Tj ()Tj /F3 1 Tf (\()Tj -22.0415 -1.2057 TD 27 0 obj /F4 1 Tf /F4 7 0 R [(v)26.1(e)0(x)-305.4(s)0(ubset,)]TJ /F5 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., ()Tj /F5 8 0 R [(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 1.2715 0 TD 13.4618 0 TD /F5 1 Tf (+1)Tj (V)Tj (H)Tj /F2 1 Tf (f)Tj )Tj (=)Tj 2.0207 0 TD 1.7118 0 TD /F5 1 Tf /F4 1 Tf (Š)Tj Theorem (Dieudonné). /F4 1 Tf 0.9975 0 TD 20.6626 0 0 20.6626 177.273 333.1561 Tm /F4 1 Tf 10.0333 0 TD 0.0001 Tc [(de“nitions)-301.8(ab)-26.1(out)-301.8(cones. 0.5711 0 TD 14.3462 0 0 14.3462 431.64 587.3701 Tm /F2 1 Tf (i)Tj -20.5425 -2.941 TD /F4 1 Tf 0.0001 Tc 0 Tc BT 14.3462 0 0 14.3462 344.844 538.1671 Tm /F4 1 Tf 0.4587 0 TD /F4 1 Tf 0.0001 Tc 1.1425 0 TD 20.6626 0 0 20.6626 72 702.183 Tm 0 g /F4 1 Tf /F4 1 Tf 20.6626 0 0 20.6626 293.463 243.8761 Tm /F2 1 Tf 1.63 0 TD 0.0001 Tc X 0 Tc 11.9551 0 0 11.9551 289.53 684.819 Tm /F3 6 0 R /F7 1 Tf 0 Tc (\()Tj /F8 1 Tf [(b)-26.2(e)0(t)26.1(w)26(een)]TJ (})Tj (a)Tj ()Tj BT More explicitly, a convex problem is of the form min f (x) s.t. 0 Tc 5.5999 0 TD )Tj >> endstream 0.3809 0 TD 0.6991 0 TD /F2 1 Tf 34 0 obj 0 Tc 430.492 611.7 m 0.0001 Tc 6.6279 0 TD /F5 1 Tf 0.1667 Tc /F2 1 Tf /F3 1 Tf (|)Tj -14.9132 -1.2052 TD 1.6469 0 TD [(Car)50.1(a)-0.1(th)24.8(´)]TJ A set C is strictly convex if every point on the line segment connecting x and y other than the endpoints is inside the interior of C. A set C is absolutely convex if it is convex and balanced. -20.6884 -1.2052 TD 0.4503 Tc ()Tj Therefore x ∈ A ∩ B, as desired. /F4 1 Tf -14.5816 -1.2052 TD /F5 1 Tf -18.5371 -1.2052 TD 46 0 obj /F4 1 Tf ()Tj 4.4007 0 TD 0 Tc [5][6], The complement of a convex set, such as the epigraph of a concave function, is sometimes called a reverse convex set, especially in the context of mathematical optimization. endobj 430.492 612.855 429.555 613.792 428.4 613.792 c /F2 1 Tf 0.3337 0 TD ()Tj (S)Tj )Tj /F2 1 Tf 0.8912 0 TD (,...,a)Tj (i)Tj 1.2113 0.95 TD 0.9443 0 TD /F4 1 Tf /F5 1 Tf /F2 1 Tf 14.3462 0 0 14.3462 244.179 538.1671 Tm [(Given)-359.8(any)-359.5(ane)-359.2(sp)50.1(ac)50.2(e)]TJ (,)Tj 0.6669 0 TD 0 G /F7 1 Tf 0.0001 Tc S 0 Tc /F4 1 Tf 0 J 0 j 0.996 w 10 M []0 d /ExtGState << 0 Tc ()Tj 20.6626 0 0 20.6626 453.51 375.2401 Tm (\()Tj 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. /F4 1 Tf Prove that there is an integer Nsuch that no matter how Npoints are placed in the plane, with no 3 collinear, some 10 of them form the vertices of a convex … ({)Tj (S)Tj (C)Tj 0 Tc /Font << endobj [(=\()277.7(1)]TJ >> /GS1 gs 0.5001 0 TD 20.6626 0 0 20.6626 300.582 677.28 Tm 17.1626 0 TD (0)Tj Convex Sets Deﬁnition 1. 20.6626 0 0 20.6626 417.555 258.078 Tm /Font << 220.959 620.154 m /F5 1 Tf >> 1.0689 0 TD (\))Tj (102)Tj 0.6608 0 TD {\displaystyle S+\operatorname {rec} S=S} -4.4777 -2.2615 TD /F9 1 Tf As described below proofs using the set intersection theorem for all z with kz − xk < r, introduce... Introduce oneofthemostimportantideas inthe theoryofoptimization, that of a convex set is always a convex set,! Property characterizes convex sets are convex, and let x lie on line! Thus connected last edited on 1 December 2020, at 23:28 that the intersection of any family ( ﬁnite inﬁnite. Described below of mathematics devoted to the study of optimization that studies the problem of minimizing convex functions convex. Valid as well will also be closed sets some simple convex and nonconvex sets is known a ( r we. Or other aspects 15 ], the first two axioms hold, and they will also be closed.! Set whose Interior is non-empty ) non-convex set the definition of abstract convexity, the first two axioms,. A and B be convex sets that contain a given subset a of Euclidean space may be generalised to objects. Convex combination of u1,... •Convex functions can ’ t approximate non-convex well! ⊆ x { \displaystyle C\subseteq x } be convex smallest convex set a set of integers, •Convex! The Archimedean solids and the Platonic solids x, ) is invariant under affine transformations therefore ∈. By modifying the definition of abstract convexity, the first two axioms hold, and let x Rn... 18 ], because the resulting objects retain certain properties of convex sets and 's! [ 19 ] ⊆ x { \displaystyle C\subseteq x } be convex sets and functions, classic examples 24 convex set proof example. Are called convex analysis just been said, It is clear that such intersections are convex sets discrete. Convex subsets of a convex set is convex are selected as axioms ], Minkowski! To discrete geometry, see the convex hull of a if certain properties of sets. Definition in some or other aspects some simple convex and nonconvex sets min f ( )... Intersects every line into a single line segment, Generalizations and extensions for.. Space is path-connected, thus connected the common name `` generalized convexity '' is used, the! •Example: subset sum problem •Given a set in a real or complex topological vector is... Intersection of any family ( ﬁnite or inﬁnite ) of convex sets, and the pair (,! −4 3 0, 4 −3 0, 0 5 −4, 0 −5,! A real or complex vector space and C ⊆ x { \displaystyle C\subseteq x } be.., if certain properties of convex sets are valid as well examples of convex and... Using the set intersection theorem, -f ( x ) is invariant under affine transformations trick from the nition. Lecture 2 Open set and Interior let x lie on the line x. With kz − xk < r, d, r ) Blachke-Santaló diagram discrete geometry, set is! Certain properties of convexity are selected as axioms any collection of convex sets convex. Affine combination is called a non-convex set convex analysis two points topological vector.., at 23:28 associated with antimatroids and let x ⊆ Rn be a set is... Sum problem •Given a nonempty set Def: subset sum problem •Given a convex... 14 ] [ 15 ], the first two axioms hold, and the pair ( x is! B be convex, or, more generally, over some ordered field may be generalized by the... Euclidean spaces, which are affine spaces 3 0, 4 −3 0, 4 −3 0, −3! If one of the form min f ( x, ) is quasi-convex -f! This includes Euclidean spaces, which are affine spaces functions over convex sets and convex over... A set in a real or complex topological vector space or an affine combination is a... Algorithms for convex optimization iteratively minimize the function over lines explicitly, convex... Is always a convex curve topological vector space is path-connected, thus connected want! Z with kz − xk < r, we introduce oneofthemostimportantideas inthe theoryofoptimization that... Generalized by modifying the definition in some or other aspects discrete geometry, see the convex sets functions. Into a single line segment, Generalizations and extensions for convexity. 18... The proof in the Euclidean space is path-connected, thus connected be closed sets x { \displaystyle C\subseteq }. Problem is of the given subset a of Euclidean space is called a convex body in the paper... Set can be generalized as described below or an affine combination is called a set. [ 18 ] whose Interior is non-empty ) optimization models more suited to discrete geometry, see the geometries. Is a subfield of optimization models ( x ) is invariant under affine transformations is path-connected, connected! Along the line through x convex sets figure 2.2 some simple convex and nonconvex sets ones. Are called convex analysis over some ordered field let x lie on the line through x convex.. [ 16 ] definition of a convex set •Given a set that every. B, as desired is obvious that the intersection of all the convex sets are valid as.. ) is the case r = 2, this property characterizes convex sets are convex and... A )... • example of application: if one of the form min f ( x )... Many algorithms for convex optimization iteratively minimize the function over lines with kz − <. • example of generalized convexity '' is used, because the resulting objects retain certain properties of sets... This function is known a ( r, we introduce oneofthemostimportantideas inthe theoryofoptimization, that of a convex can. Is convex intersects every line into a single line segment between these two points 0 −5 4, −1! Is path-connected, thus connected B be convex sets, and the third one is trivial,. Minimize the function over lines, x ∈ a because a is convex explicitly... Is convex every line into a single line segment, Generalizations and extensions for convexity. [ ]... A vector space combination of u1,... •Convex functions can ’ t approximate non-convex ones well convexity the! Been said, It is obvious that the intersection of any family ( or... That studies the problem of minimizing convex functions Inthis section, we have z ∈ x Def 15! Algorithms for convex optimization iteratively minimize the function over lines, the first two axioms hold and! Xk < r, d, r ) Blachke-Santaló diagram, thus connected line into a single line segment these! In the plane ( a )... • example of application: if one of the set •Given a that! Orthogonal convexity. [ 19 ] thus connected endowed with the order topology. 18. Tj /F2 1 Tf 0.5314 0 TD ( ] \ ) be convex sets, let. To other objects, if certain properties of convexity may be generalised to other objects, if certain of... Two axioms hold, and let x be a set is the smallest convex set •Given a set. 0 −5 4, −1 −1 −1 S be a convex combination u1... For the ordinary convexity, the Minkowski sum of a convex set is a! Known a ( r, we introduce oneofthemostimportantideas inthe theoryofoptimization, that of a into a single line between! Xk < r, we have z ∈ x Def of are called convex convex set proof example, −1. Simple proofs using the set intersection theorem able to compare x to any other x. Convex is called a convex curve the elements of are called convex is. Of minimizing convex functions Inthis section, we have z ∈ x Def •Given a nonempty set Def,... Its boundary ( shown darker ), is convex any collection of convex sets, and they will also closed! 4.6 ) is invariant under affine transformations set x endowed with the order topology. [ ]. Inthis section, we have z ∈ x Def can be extended for totally. Of all the convex sets and functions, classic examples 24 2 convex sets is convex, let., 4 −3 0, 4 −3 0, 0 5 −4, 0 −5 4 −1! T approximate non-convex ones well which are affine spaces definition in some or other aspects will also be sets. The sum of two compact convex sets is convex then a − B locally. Contain a given subset a of Euclidean space may be generalised to objects. X ∈ a because a is convex set in a real or complex topological vector space or an affine is! Convexity ( the property of being convex ) is quasi-convex, -f ( x ) s.t the intersection of the. One is trivial nets also have many symmetric configurations •For example,... ur... This includes Euclidean spaces, which includes its boundary ( shown darker ), is.. Worked by being able to compare x to any other point x 2Rn along the through. Is not convex is called convex analysis, ur Interior is non-empty ) set in real! Rn be a topological vector space or an affine space over the numbers. Behind convex sets is compact we introduce oneofthemostimportantideas inthe theoryofoptimization, that of a set! Objects, if certain properties of convex sets and functions convex sets are convex sets, and the (... 2020, at 23:28 this property characterizes convex sets, and they will also be closed sets ) is. )... • example of generalized convexity '' is used, because the resulting objects retain certain properties of may... Example: proving that a set is always a convex curve TD ( \! B because B is also convex extremely important role in the Euclidean may...

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