熵正則化及無窮多個比值的廣義型分數規劃問題

行政院國家科學委員會專題研究計畫 成果報告

熵正則化及無窮多個比值的廣義型分數規劃問題

計畫類別： 個別型計畫 計畫編號： NSC91-2115-M-006-017- 執行期間： 91 年 08 月 01 日至 92 年 10 月 31 日 執行單位： 國立成功大學數學系暨應用數學所 計畫主持人： 許瑞麟 報告類型： 精簡報告 報告附件： 出席國際會議研究心得報告及發表論文 處理方式： 本計畫可公開查詢

中 華 民 國 92 年 10 月 29 日

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Crouzeix, Ferland and

Schaible [1]

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(P)

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(P_α) P (α) = min

x∈Xmaxt∈T [ft(x)− αgt(x)].

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Example: Let X = [1/2, 1]× [1/2, 1] and T = [0, ∞).

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: inf (x1,x2)∈X sup t∈T{(−2x1− 1)/(tx1+ x2+ 1)}.

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α∗ _{= 0.}

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P (α) = inf (x1,x2)∈X sup t∈T{−2x1− 1 − α(tx1+ x2+ 1)},

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úk

α > 0, P (α) = −2α − 3.

úk

α = 0, P (α) = −3.

úk

α < 0, P (α) = inf (x1,x2)∈X∞ = ∞.

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P (α∗) = P (0) =−3.

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P (α)

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(1) Theorem If both X and T are compact, then P (α) < 0 if and only if α > α∗.

(2) Theorem If X and T are compact, then P (α∗) = 0. Moreover, problem (P)

and (P_α∗) have the same optimal solutions, that is, argmin

x∈X maxt∈T {ft(x)/gt(x)} = argminx∈X maxt∈T {ft(x)− α ∗_g

t(x)}.

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Iterative Entropic Regularization [2]

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Dinkelbach

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Modified Dinkelbach algorithm with entropic regularization,

˚Ñ

MDER.

2 Algorithm (MDER)

Step 1. Given x₀∈ X, t₁ ∈ T and η, γ > 0, 0 < δ < 1. Compute ˜

α₁ = max

t∈T {ft(x0)/gt(x0)}.

Set p = 10, m = 1, T₁={t₁}, k = 1, j = 1 and ˜xm_{p, ˜α}

k = x0.

Step 2. Approximate the following continuous minimax problem (P_˜αk) P (˜α_k) = min

x∈Xmaxt∈T [ft(x)− ˜αkgt(x)].

by the iterative entropic regularization [2]

F_{p, ˜α}m k(x) = (1/p) ln
_m  i=1 exp(p(f_ti(x)− ˜α_kg_ti(x)))  . (1) Find ˜xm_{p, ˜α} k that satisfies F_{p, ˜α}m _k(˜xm_{p, ˜αk})≤ min x∈XF m p, ˜αk(x) + δj (2) and compute ˜ r_k = max t∈T {ft(˜x m p, ˜α_k)− ˜αkgt(˜xm_{p, ˜α} k)}. (3)

Step 3. If ˜r_k < −η, update the Dinkelbach parameter ˜ α_k+1 = max t∈T {ft(˜x m p, ˜α_k)/gt(˜xm_{p, ˜α} k)}; (4)

the iteration counter k = k + 1; and η = η∗ (1 + δ). Go to Step 2.

Step 4. If ˜r_k ≥ −η, set η = |˜r_k|; improve m and p to obtain a better approximate solution. Step 4.1. If ˜r_k > F_{p, ˜α}m k(˜xmp, ˜αk) + δk, choose t_m+1 ∈ argmax t∈T {ft(˜x m p, ˜α_k)− ˜αkgt(˜xmp, ˜α_k)}. (5)

Set T_m+1= T_m∪ {t_m+1}, p = p1+ε and j = j + 1. Go to Step 2. Step 4.2. δj+ (ln m)/p > γ− δk, set p = p1+ε and j = j + 1. Go to Step 2. Step 5. (Stopping Criteria)

If η > γ, compute ˜α_k+1 = max

t∈T {ft(˜x m

p, ˜α_k)/gt(˜xm_{p, ˜α}

k)}, k=k+1 and return to

Step 2. Otherwise, stop and report that ˜xm_{p, ˜α}

k is a γ-optimal solution of (P).

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MDER

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(3) Lemma Let α∗ be the optimal value of problem (P). Then, ˜α_k ≥ α∗.

(4) Lemma The optimal value P ( ˜α_k) of parametric problem P_˜αk is non-positive.

(5) Lemma F_{p, ˜α}m

k(x∗˜αk)≤ max_t∈Tm[ft(x∗˜αk)− ˜αkgt(x∗˜αk)] + (ln m)/p

(6) Lemma If ˜r_k ≤ F_{p, ˜α}m

k(˜xmp, ˜αk) + δk, then

(7) Lemma If max{−η, P (˜α_k)} ≤ ˜r_k ≤ P (˜α_k) + (ln m)/p + δj+ δk, then −δk− δj− (ln m)/p − η ≤ P (˜αk)≤ 0.

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Step5

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, P ( ˜α_k)

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0

íÏÏBÖ

u

2γ,

6ÿuBbªJ)ƒø_¸ˇÊ

2γ

í¡Nj

(2γ-optimal solution).

(8) Theorem When the algorithm stops in Step 5 with k = ¯k, then the following inequality

−2γ ≤ P (˜α_¯k)≤ 0

hold.

(9) Lemma If the algorithm does not stop at Step 5, then

˜

r_k < −γ < 0 and

−γ + ˜rk ≤ P (˜αk).

(10) Lemma {˜α_k} is a decreasing sequence.

(11) Lemma If ˜α_k > ˜α_k+1, then P ( ˜α_k)≤ P (˜α_k+1) + ( ˜α_k+1− ˜α_k) min

t∈T gt(x ∗

˜αk+1).

(12) Lemma If ˜α_k > α∗, then P ( ˜α_k)≤ P (α∗) + (α∗− ˜α_k) min

t∈T gt(x ∗_). (13) Theorem If P ( ˜α_k) < ˜r_k < −γ < 0, then ( ˜α_k+1− α∗)/( ˜α_k− α∗)≤ 1 − m(g)/M(g) − ˜r_k/( ˜α_k − α∗) max t∈T gt(˜x m p, ˜α_k)  .

(14) Condition There exists some positive integer K such that

0 < ε_k < (m(g)/M(g)) − γ, ∀k > K.

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Ê_çí‘K-

(Condition 14),

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MDER

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(15) Theorem Under Condition 14, if the algorithm does not stop at Step 5, then, from certain iterate K on,

( ˜α_k+1− α∗)/( ˜α_k − α∗)≤ 1 − γ, ∀k > K

In other words, {˜α_k} converges to α∗ linearly.

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Example1.

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min

x∈Xmaxt∈T {(−tx1+ x2)/(x1+ x2)}

w2

X = [1/2, 30] × [1/2, 100]

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T = [0, 1].

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l)ƒu

1/61,

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(30, 1/2).

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Table 1.

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Table 1 Computational results for Example 1

x₀ δ γ Iters. x∗ Optvalue Time

(1/2, 100) 0.25 1e-5 38 (30, 0.5) 0.016393 2.016

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Example 1

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4 Example 2.

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min x∈Xmaxt∈T {(t 2_x 1x2+ x2t1 + tx33)/(5(t− 1)2x41+ 2x22+ 4tx3)}

w2

(x₁, x₂, x₃)∈ X = [1/2, 5] × [1/2, 5] × [1/2, 5]

¸

t ∈ T = [0, 1].

ƶ

MDER

à

(5, 5, 5)

ç–áõ1/!!Ê

(0.5, 1.4916, 0.5),

7|7MÑ

−0.209954.

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P (−0.209954) = min x∈Xmaxt∈T {f(t) + 0.209954g(t)} = 0

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Theorem 1

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2

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(0.5, 1.4916, 0.5)

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Example 2

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Table 2

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Table 3.

Table 2 Computational results for Example 2

x₀ γ Iters. x∗ Optimal value Time

(5,5,5) 1e-2 112 (0.5, 1.491666160, 0.5) -0.209954624 5.703 Table 3 Verification of optimal value -0.209954 by Matlab

α P (α)

-0.209554 -3.4010e-005

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Table 5.4

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[1] J. P. Crouzeix, J. A. Ferland, and S. Schaible. An algorithm for generalized fractional programs.

Journal of Optimization Theory and Applications 47 (1985), 35–49.

[2] R. L. Sheu and J. Y. Lin. On solving continuous min-max problems by iteratively entropic-regularization method. To appear in Journal of Optimization Theory and Applications (2003).