Convex Optimization. Mikhail Moklyachuk

on the circle , and hence in X, then the value of f will decrease. Consequently, these points are not points of the local minimum f on X. At the same time, for any ε > 0, the point (–ε, 1) belongs to X and f(0, 1) < f(–ε, 1). Therefore, the point (0, 1) is not a point of the local maximum f on X. Consequently, the stationary points (0, 1) and (0, –1) are not solutions of the problem.

      We now consider the matrix of the second derivatives of the Lagrangian function:

      For values with (5), this matrix is as follows:

      Since λ₀ < 0, this matrix is positive definite. Sufficient conditions for the minimum are fulfilled. Consequently, (, ) is the point of the strict local minimum of f on X.

      Answer. ∈ absmin, ∈ absmin, ∈ absmax. Δ

1.5. Exercises

      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)