Читать книгу Convex Optimization - Mikhail Moklyachuk - Страница 13

Let us solve the following optimization problems.

1 1) f(x, y) = x4 + y4 − 4xy → extr.

2 2) f(x, y) = ae−x + be−y + ln(ex + ey) → extr.

3 3) f(x, y) = (x + y)(x − a)(y − b) → extr.

4 4) f(x, y) = x2 − 2xy2 + y4 − y5 → extr.

5 5) f(x, y) = x + y + 4 sin (x) sin (y) → extr.

6 6) f(x, y) = xex − (1 + ex) cos (y) → extr.

7 7) f(x, y) = (x2 + y2)e−(x2 + y2) → extr.

8 8) f(x, y) = xy ln (x2 + y2) → extr.

9 9) .

10 10) f(x, y) = sin (x) sin (y) sin(x + y) → extr, 0 ≤ x ≤ π, 0 ≤ y ≤ π.

11 11) f(x, y) = sin (x) +cos (y) +cos (x − y) → extr, 0 ≤ x ≤ π/2, 0 ≤ y ≤ π/2.

12 12) f(x, y) = x2 + xy + y2 − 4 ln (x) − 10 ln (y) → extr.

13 13) f(x, y) = (5x + 7y − 25)e−(x2 + y2 + xy) → extr.

14 14) f(x, y) = ex2−y (5 − 2x + y) → extr.

15 15) f(x, y) = e2x+3y(8x2 − 6xy + 3y2) → extr.

16 16) .

17 17) .

18 18) .

19 19) f(x, y) = 2x4 + y4 − x2 − 2y2 → extr.

20 20) f(x, y) = x2 − xy + y2 − 2x + y → extr.

21 21) f(x, y) = xy + 50/x + 20/y → extr.

22 22) f(x, y) = x2 − y2 − 4x + 6y → extr.

23 23) f(x, y) = 5x2 + 4xy + y2 − 16x − 12y → extr.

24 24) f(x, y) = 3x2 + 4xy + y2 − 8x − 12y → extr.

25 25) f(x, y) = 3xy − x2y − xy2 → extr.

26 26) f(x, y, z) = x2 + y2 + z2 − xy + x − 2z → extr.

27 27) f(x, y, z) = x2 + 2y2 + 5z2 − 2xy − 4yz − 2z → extr.

28 28) f(x, y, z) = xy2z3(a − x − 2y − 3z) → extr, a > 0.

29 29) f(x, y, z) = x3 + y2 + z2 + 12xy + 2z → extr, x > 0, y > 0, z > 0.

30 30) f(x, y, z) = x + y2/4x + z2/y + 2/z → extr.

31 31) f(x, y, z) = x2 + y2 + z2 + 2x + 4y − 6z → extr.

32 32) f(x, y) = y → extr, x3 + y3 − 3xy = 0.

33 33) f(x, y) = x3 + y3 → extr, ax + by = 1, a > 0, b > 0.

34 34) f(x, y) = x3/3 + y → extr, x2 + y2 = a, a > 0.

35 35) f(x, y) = x sin (y) → extr, 3x2 − 4 cos (y) = 1.

36 36) f(x, y) = x/a + y/b → extr, x2 + y2 = 1.

37 37) f(x, y) = x2 + y2 → extr, x/a + y/b = 1.

38 38) f(x, y) = Ax2 + 2Bxy + Cy2 → extr, x2 + y2 = 1.

39 39) f(x, y) = x2 + 12xy + 2y2 → extr, 4x2 + y2 = 25.

40 40) f(x, y) = cos2 (x) + cos2 (y) → extr, x − y = π/4.

41 41) f(x, y) = x/2 + y/3 → extr, x2 + y2 = 1.

42 42) f(x, y) = x2 + y2 → extr, 3x + 4y = 1.

43 43) f(x, y) = exy → extr, x + y = 1.

44 44) f(x, y) = 5x2 + 4xy + y2 → extr, x + y = 1.

45 45) f(x, y) = 3x2 + 4xy + y2 → extr, x + y = 1.

46 46) f(x, y, z) = xy2z3 → extr, x + y + z = 1.

47 47) f(x, y, z) = xyz → extr, x2 + y2 + z2 = 1, x + y + z = 0.

48 48) f(x, y, z) = a2x2 + b2y2 + c2z2 − (ax2 + by2 + cz2)2 → extr, x2 + y2 + z2 = 1, a > b > c > 0.

49 49) f(x, y, z) = x + y + z2 + 2(xy + yz + zx) → extr,x2 + y2 + z = 1.

50 50) f(x, y, z) = x − 2y + 2z → extr, x2 + y2 + z2 = 1.

51 51) f(x, y, z) = xm yn zp → extr, x + y + z = a, m > 0, n > 0, p > 0, a > 0.

52 52) f(x, y, z) = x2 + y2 + z2 → extr, x2/a2 + y2/b2 + z2/c2 = 1, a > b > c > 0.

53 53) f(x, y, z) = xy2z3 → extr, x + 2y + 3z = a, x > 0, y > 0, z > 0, a > 0.

54 54) f(x, y, z) = xy + yz → extr, x2 + y2 = 2, y + z = 2, x > 0, y > 0, z > 0.

55 55) f(x, y, z) = sin (x) sin (y) sin(z) → extr, x + y + z = π/2.

56 56) f(x, y) = ex−y − x − y → extr, x + y ≤ 1, x ≥ 0, y ≥ 0.

57 57) f(x, y) = x2 + y2 − 2x − 4y → extr, 2x + 3y − 6 ≤ 0, x + 4y − 5 ≤ 0.

58 58) f(x, y) = 2xy − x2 − 2y2 → extr, x − y + 1 ≥ 0, 2x + 3y + 6 ≤ 0.

59 59) f(x, y) = x2 + y2 → extr, −5x + 4y ≤ 0, −x + 4y + 3 ≤ 0.

60 60) f(x, y) = x2 + y2 − 2x → extr, x − 2y + 2 ≤ 0, 2x − y ≥ 0.

61 61) f(x, y, z) = xyz → extr, x2 + y2 + z2 ≤ 1.

62 62) f(x, y, z) = 2x2 + 2x + 4y −3z → extr, 8x −3y + 3z ≤ 40,− 2x + y −z = −3, y ≥ 0.

63 63) f(x, y, z) = x2 + 4y2 + z2 → extr, x + y + z ≤ 12, x ≥ 0, y ≥ 0, z ≥ 0.

64 64) f(x, y, z) = 3y2 − 11x − 3y −z → extr, x − 7y + 3z + 7 ≤ 0, 5x + 2y −z ≤ 2, z ≥ 0.

65 65) f(x, y, z) = xz − 2y → extr, 2x − y − 3z ≤ 10, 3x + 2y + z = 6, y ≥ 0.

66 66) f(x, y, z) = −4x − y + z2 → extr, x2 + y2 + xz − 1 ≤ 0, x2 + y2 − 2y ≤ 0, 5 − x + y + z ≤ 0, x ≥ 0, y ≥ 0, z ≥ 0.

67 67) , , b > 0, xj ≥ 0, αj > 0, βj > 0, aj > 0, j = 1, 2, … , n.

68 68) , , b > 0, xj ≥ 0, cj > 0, αj > 0, βj > 0, j = 1, 2, … , n.

69 69) , , b > 0, xj > 0, cj > 0, αj > 0, βj > 0, j = 1, 2, … , n.

70 70) , b > 0, α > 0, xj > 0, cj > 0, j = 1, 2, … , n.

71 71) , , b > 0, 0 < α < 1, xj > 0, cj > 0, j = 1, 2, … , n.

72 72) , , cj > 0, a = (a1, … , an) ≠ 0, α = 2m, m ∈ N.

73 73) , , cj > 0, a = (a1, … , an) ≠ 0, α > 1, b > 0.

74 74) , , b > 0, aj > 0, c = (c1, … , cn) ≠ 0, α = 2m, m ∈ N.

75 75) , , b > 0, aj > 0, c = (c1, … , cn) ≠ 0, α > 1.

76 76) , , b > 0, c = (c1, … , cn) ≠ 0, α > 1.

77 77) Divide the number 8 into two parts so that the product of their product on the difference is maximal (Niccolo Tartaglia problem).

78 78) Determine the rectangular triangle of the largest area, provided that the sum of the lengths of its legs is equal to a given number (Fermat’s problem).

79 79) On the BC side of a triangle ABC, define a point E such that the parallelogram ADEK, whose points D and K lie on sides AB and AC, respectively, has the largest area (Euclid problem).

80 80) On a given face of a tetrahedron, take a point through which planes parallel to three other faces are drawn. Choose a point so that the volume of the parallelepiped is maximal (generalized Euclid problem).

81 81) Determine a polynomial of the second-degree t2 + x1t + x2 such that the integralgets the smallest value (Legendre problem for polynomial of the second degree).

82 82) Determine a polynomial of the third degree t3 + x1t2 + x2t + x3 such that the integralgets the smallest value (Legendre problem for polynomial of the third degree).

83 83) Among all discrete random variables that take n values, determine a random variable with the largest entropy. The entropy of the sequence of positive numbers p1, … , pn, such that , is the number

84 84) Insert a rectangle of maximum area into a circle.

85 85) Insert a triangle of maximum area into a circle.

86 86) Insert a cylinder with maximum volume into a ball (Kepler’s problem).

87 87) Insert a cone with maximum volume into a ball.

88 88) Among cones inscribed in a ball, determine a cone with the maximum area of the lateral surface.

89 89) Insert in a sphere from the space ℝn a rectangular parallelepiped with the largest volume.

90 90) Insert a tetrahedron with the largest volume into a ball.

91 91) Among triangles with a given perimeter determine a triangle of the largest area.

92 92) Among all n-angles of a given perimeter determine an n-cube of the largest area (Zeno’s problem).

93 93) Insert n-angles of the maximum area in a circle.

94 94) On the diameter AB of a circle of the unit radius, a point E is taken through which a chord CD is drawn. Determine a position of the chord in which the square of the quadrilateral ABCD is maximal.

95 95) Determine in a triangle such a point that the sum of the ratio of lengths of sides and distances from the point to relevant sides is minimal.

96 96) Insert into a circle a triangle with the largest sum of squares of sides.

97 97) Through a given point inside a corner, draw a segment with ends on the sides of the corner so that the area of the formed triangle is minimal.

98 98) Through a point inside a corner draw a section with ends on the sides of the corner so that the perimeter of the formed triangle is minimal.

99 99) Determine a quadrilateral with given sides of the largest area.

100 100) Among segments of a ball having a given area of the lateral surface, find the segment with the largest volume (Archimedes’ problem).

101 101) Determine a point C on a line such that the sum of the distances from the point C to the given points A and B is minimal.

102 102) Among all tetrahedra with a given base and height, find a tetrahedron with the smallest lateral surface.

103 103) Among all tetrahedra with a given base and area of the lateral surface, find a tetrahedron with the largest volume.

104 104) Among all tetrahedra having a given area of the surface, find a tetrahedron that has the largest volume.

105 105) Three points x1, x2, x3 are given on a plane. Determine a point x0 such that the sum of the squares of distances from the point x0 to the points x1, x2, x3 is the smallest.

106 106) In the space ℝn there are given N points x1, … , xN and N positive numbers m1, … , mN. Determine a point x0, such that the sum with the coefficients mi of the squares of distances from the point x0 to X1, … , xN is the smallest.

107 107) Solve the previous problem, provided that the point x0 lies on the sphere of unit radius.

108 108) Solve the previous problem, provided that the point x0 belongs to the ball of unit radius.

109 109) Find the distance from a point to the ellipse. How many normals can be drawn from a point to the ellipse (Apollonius’s problem)?

110 110) Find the distance from a point x0 to a parabola.

111 111) Find the distance from a point x0 to a hyperbole.

112 112) Find the distance from a point x0 in the space ℝn to the hyperplane H = {x ∈ ℝn|〈a, x〉 = β}.

113 113) Find the distance from a point x0 to the hyperplane in a Hilbert space.

114 114) Find the distance from a point x0 in the space ℝn to a line.

115 115) Find the minimum of a linear function in the space ℝn on a unit ball.

116 116) In the ellipse x2/a2 + y2/b2 = 1 insert a rectangle of the largest area with sides parallel to the coordinate axes.

117 117) In the ellipsoid x2/a2 + y2/b2 + z2/c2 = 1 insert a rectangular parallelepiped of the largest volume with edges parallel to the axes of coordinates.

118 118) Prove the inequality between the power averagessolving the problem

119 119) Prove the inequality

120 120) Prove the inequality

121 121) Prove the Hölder inequalityMake sure that for y = (y1, … , yn) = 0, the equality holds only when |xi| = λ|yi|, i = 1, … , n.

122 122) Prove the Minkowski inequalityMake sure that for y = (y1, … , yn) = 0, the equality holds only when xi = λyi, λ > 0,i = 1, … , n.