A Spline Smoothing Newton Method for Semi-Infinite Minimax Problems

Abstract

Based on discretization methods for solving semi-infinite programming problems, this paper presents a spline smoothing Newton method for semi-infinite minimax problems. The spline smoothing technique uses a smooth cubic spline instead of max function and only few components in the max function are computed; that is, it introduces an active set technique, so it is more efficient for solving large-scale minimax problems arising from the discretization of semi-infinite minimax problems. Numerical tests show that the new method is very efficient.

1. Introduction

In this paper, we consider the following semi-infinite minimax problems: (𝑃)min_{𝑥∈𝑅𝑛}max_𝑦∈𝑌𝜓𝑥,𝑦, (1) where 𝜓∶𝑅^𝑛×𝑅^𝑚→𝑅. We assume (as in Assumption  3.4.1 in [1]) that both 𝜓(⋅,⋅) and ∇_𝑥𝜓(⋅,⋅) are Lipschtiz continuous on bounded sets. Such a semi-infinite minimax problem 𝑃 is an exciting part of mathematical programming. It has very widespread application backgrounds in optimal electronic circuit design, linear Chebyshev approximation, minimization of floor area, optimal control, computer-aided design, numerous engineering problems, and so forth (see [1–8]). Over the past decade, many researchers had done a lot of works on it and proposed some algorithms (see [9–14]). However, efficient algorithms for solving the problem 𝑃 are few, because it is difficult to design an algorithm to deal with the nondifferentiability of objective function and the infinite set 𝑌. A common approach for solving 𝑃 is the discretization method. Generally, discretization of multidimensional domains gives rise to minimax problems with thousands of component functions. Computation cost is increased; efficiency of the discretization method is affected. To overcome these problems, Polak et al. proposed algorithms with smoothing techniques for solving finite and semi-infinite minimax problems (see [15, 16]). In [16], an active set strategy which can be used in conjunction with exponential (entropic) smoothing for solving large-scale minimax problems arising from the discretization of semi-infinite minimax problems had been proposed. But the active set grew monotonically. In this paper, using the feedback precision-adjustment smoothing parameter rule which was proposed by Polak et al. in [15], we propose a new discretization smoothing algorithm for solving 𝑃. The smoothing function is cubic spline function not exponential function in our algorithm. The spline smoothing technique uses a smooth spline function instead of thousands of component functions and acts also as an active set technique, so only few components in the max function are computed at each iteration. And the active set does not grow monotonically; hence the number of gradients and Hessian calculations is dramatically reduced, and the computation cost is greatly reduced. Numerical tests show that the new method is very efficient for semi-infinite minimax problems with complicated component functions.

We assume that the problem 𝑃 can be approximated by a sequence of finite minimax problems of the form 𝑃_𝑌𝑁min_{𝑥∈𝑅𝑛}max_{𝑦∈𝑌𝑁}𝜓𝑥,𝑦, (2) where the sets 𝑌_𝑁, 𝑁=1,2,3,… have finite cardinality. In practice, we can expect that 𝑌_𝑁⊂𝑌_𝑁+1, that is, that they grow monotonically, and that the closure of ∪^∞_𝑁=1𝑌_𝑁 is equal to 𝑌. However, it suffices to assume, as in Assumption  3.4.2 in [1].

Assumption 1. There exist a strictly positive valued, strictly monotone decreasing function Θ∶𝑁→𝑅 and constants 𝐾>0 and 𝑁₀>0, such that for every 𝑁>𝑁₀ and 𝑦∈𝑌, there exists a 𝑦^′∈𝑌_𝑁 such that 𝑦−𝑦^′≤𝐾Θ(𝑁). (3)
It is easy to satisify with Assumption 1. For example, let 𝑌=[𝑎,𝑏],𝑌_𝑁=[𝑎,𝑎+(𝑏−𝑎)/2^𝑁,𝑎+2((𝑏−𝑎)/2^𝑁),…,𝑎+(2^𝑁−1)((𝑏−𝑎)/2^𝑁),𝑏],Θ(𝑁)=1/(2^𝑁−1),𝐾=1/2‖𝑌‖=(1/2)(𝑏−𝑎).
We assume that 𝑌_𝑁={𝑦_𝑁1,…,𝑦_𝑁𝑞𝑁}.
Next, we let 𝑓^𝑁𝑗(𝑥)≜𝜓𝑥,𝑦_𝑁𝑗,𝑗∈𝑄_𝑁=1,2,…,𝑞_𝑁, (4) and define 𝜙(𝑥)=max_𝑦∈𝑌𝜓,𝜙𝑥,𝑦_𝑁(𝑥)=max_{𝑗∈𝑄𝑁}𝑓^𝑁𝑗(𝑥), (5) and with this notation, the problems 𝑃_𝑌𝑁 assume the form 𝑃_𝑁min_{𝑥∈𝑅𝑛}𝜙_𝑁(𝑥). (6)
The optimality functions 𝜌_𝑁 for the problems 𝑃_𝑁 are defined by 𝜌_𝑁(𝑥)≜−min_{𝜆∈Σ𝑗∈𝑄𝑁}𝜆^𝑗𝜙_𝑁(𝑥)−𝑓^𝑁𝑗(𝑥+1)2_{𝑗∈𝑄𝑁}𝜆^𝑗∇𝑓^𝑁𝑗(𝑥)², (7) where Σ≜{𝜆∈𝑅^𝑄𝑁∣𝜆^𝑗≥0,∑_{𝑗∈𝑄𝑁}𝜆^𝑗=1}.
In [1], we find the following result.

Theorem 2. Suppose that 𝑥 is an optimal solution to problems 𝑃_𝑁; then, 𝜌_𝑁(𝑥)=0.

The corresponding optimality function for 𝑃 is defined (see Theorem  3.1.6 in [1]) by 𝜌(𝑥)=−min_{𝜂∈𝐹𝜙(𝑥)}𝜂⁰+12𝜂², (8) where 𝐹𝜙(𝑥)=conv_𝑦∈𝑌𝜙(𝑥)∇−𝜓𝑥,𝑦_𝑥𝜓.𝑥,𝑦 (9)

Referring to Lemma  3.4.3 in [1], we see that, under Assumption 1, the following result must hold.

Theorem 3. Suppose that {𝑥_𝑁}^∞_𝑁=0 is a sequence in 𝑅^𝑛 converging to a point 𝑥. Then, 𝜙_𝑁(𝑥_𝑁)→𝜙(𝑥) and 𝜌_𝑁(𝑥_𝑁)→𝜌(𝑥) as 𝑁→∞.

In this paper, we consider to approximate uniformly 𝜙_𝑁(𝑥) by the smooth spline introduced in [17].

Let us first recall the formulation of multivariate spline. Let 𝐷 be a polyhedral domain of 𝑅^𝑞𝑁 which is partitioned with irreducible algebraic surfaces into cells Δ={Δ_𝑖∣𝑖=1,…,𝑁}. A function 𝑠(𝑧) defined on 𝐷 is called a 𝑘-spline function with 𝑟th order smoothness, expressed for short as 𝑠(𝑧)∈𝑆^𝑟_𝑘(𝐷,Δ), if 𝑠(𝑧)∈𝐶^𝑟(𝐷) and 𝑠(𝑧)|_Δ𝑖=𝑝_𝑖∈𝑃_𝑘, where 𝑃_𝑘 is the set of all polynomials of degree 𝑘 or less in 𝑞_𝑁 variables. Similar to the smooth spline which approximates uniformly min{𝑧₁,𝑧₂,…,𝑧_𝑞𝑁} given in [17], we can construct a spline function 𝑠²₃(𝑧;𝜀)∈𝑆²₃(𝑅^𝑞𝑁,Δ²_MS) to approximate uniformly max{𝑧₁,𝑧₂,…,𝑧_𝑞𝑁} (as 𝜀→+0), where Δ²_MS is the homogenous Morgan-Scott partition of type two in [17], as follows: 𝑠²₃𝑧₁,𝑧₂,…,𝑧_𝑞𝑁;𝜀=𝑧_𝑖1+^𝑘−1_𝑙=1𝑐_𝑙𝑙𝑧_𝑖𝑙+1−^𝑙_𝑗=1𝑧_𝑖𝑗+𝜀³,for𝑧∈Δ_{𝑖1,…,𝑖𝑘}(𝜀), (10) where 𝑐₁=1/(6𝜀²),𝑐_𝑘/𝑐_𝑘+1=(𝑘+2)/𝑘, 1≤𝑘≤𝑞_𝑁, and the cell Δ_{𝑖1,…,𝑖𝑘}(𝜀) is the region defined by the following inequalities: 𝑧_𝑖𝑙−𝑧_𝑖𝑙+1()𝑧≥0,when1≤𝑙<𝑘,𝑘−1_𝑖𝑘−^𝑘−1_𝑗=1𝑧_𝑖𝑗+𝜀≥0,𝑘𝑧_𝑖𝑙−^𝑘_𝑗=1𝑧_𝑖𝑗+𝜀≤0,when𝑘+1≤𝑙≤𝑞_𝑁. (11) The composite function 𝛾_{𝑞𝑁,𝑡𝑁}(𝑥) approximates uniformly 𝜙_𝑁(𝑥) as 𝑡_𝑁→+0, where 𝛾_{𝑞𝑁,𝑡𝑁}(𝑥)=𝑠²₃𝑓¹(𝑥),𝑓²(𝑥),…,𝑓^𝑞𝑁(𝑥);𝑡_𝑁,𝑓for𝑥suchthat₁(𝑥),𝑓²(𝑥),…,𝑓^𝑞𝑁(𝑥)∈Δ_{𝑖1,…,𝑖𝑘}𝑡_𝑁. (12)

Proposition 4. Suppose that 𝑡_𝑁>0. For any 𝑥∈𝑅^𝑛, we define Ω(𝑥)≜𝑗∈𝑄_𝑁∣𝑓^𝑁𝑗(𝑥)=𝜙_𝑁(𝑥). (13) If the function 𝑓^𝑗(⋅)is continuous, then 𝛾_{𝑞𝑁,𝑡𝑁}(⋅) is continuous and is increasing with respect to 𝑡_𝑁. Furthermore, 𝜙_𝑁(𝑥)+𝑡_𝑁311−Ω(𝑥)≤𝛾_{𝑞𝑁,𝑡𝑁}(𝑥)≤𝜙_𝑁(𝑥)+𝑡_𝑁311−𝑘, (14) where |Ω(𝑥)| denotes the cardinality of Ω(𝑥).

Proof. 𝑠²₃(𝑧₁,𝑧₂,…,𝑧_𝑞𝑁;𝜀) is twice continuously differentiable and if the functions 𝑓^𝑗(⋅),𝑗∈𝑄_𝑁 are continuous, it is easy to know that 𝛾_{𝑞𝑁,𝑡𝑁}(⋅) is continuous. From Lemma  1.1 in [18], we know 𝛾_{𝑞𝑁,𝑡𝑁}(⋅) is increasing with respect to 𝑡_𝑁.
According to (12), we have 𝛾_{𝑞𝑁,𝑡𝑁}(𝑥)=𝑓^𝑁𝑖1(𝑥)+𝑐₁𝑓^𝑁𝑖1(𝑥)−𝑓^𝑁𝑖2(𝑥)+𝑡_𝑁³+⋯+𝑐_|Ω(𝑥)|Ω(𝑥)𝑓^{𝑁𝑖|Ω(𝑥)|+1}(𝑥)−𝑓^𝑁𝑖1(𝑥)+⋯+𝑓^{𝑁𝑖|Ω(𝑥)|}(𝑥)+𝑡_𝑁³+⋯+𝑐_𝑘−1(𝑘−1)𝑓^𝑁𝑖𝑘(𝑥)−𝑓^𝑁𝑖1(𝑥)+⋯+𝑓^{𝑁𝑖𝑘−1}(𝑥)+𝑡_𝑁³. (15) From the definition of Ω(𝑥), we know (𝑙−1)𝑓^𝑁𝑖𝑙(𝑥)−(𝑓^𝑁𝑖1(𝑥)+⋯+𝑓^{𝑁𝑖𝑙−1}(𝑥))+𝑡_𝑁=𝑡_𝑁,1≤𝑙≤|Ω(𝑥)|. From the definition of Δ_{𝑖1,…,𝑖𝑘}(𝑡_𝑁), we know 𝑡_𝑁≥(𝑙−1)𝑓^𝑁𝑖𝑙(𝑥)−(𝑓^𝑁𝑖1(𝑥)+⋯+𝑓^{𝑁𝑖𝑙−1}(𝑥))+𝑡_𝑁≥0,1≤𝑙<𝑘. Then, we have 𝑓^𝑁𝑖1(𝑥)+^{|Ω(𝑥)|−1}_𝑙=1𝑐_𝑙𝑡³_𝑁≤𝛾_{𝑞𝑁,𝑡𝑁}(𝑥)≤𝑓^𝑁𝑖1(𝑥)+^𝑘−1_𝑙=1𝑐_𝑙𝑡³_𝑁. (16) By 𝑐_𝑙/𝑐_𝑙+1=(𝑙+2)/𝑙, we can obtain 𝑐_𝑙=(2/(𝑙(𝑙+1)))𝑐₁=1/(3𝑙(𝑙+1)𝑡²_𝑁). Thus ∑^{|Ω(𝑥)|−1}_𝑙=1𝑐_𝑙𝑡³_𝑁=(𝑡_𝑁/3)(1−(1/|Ω(𝑥)|)) and ∑^𝑘−1_𝑙=1𝑐_𝑙𝑡³_𝑁=(𝑡_𝑁/3)(1−(1/𝑘)) and 𝑓^𝑁𝑖1(𝑥)=𝜙_𝑁(𝑥). Hence the desired result follows.

The following proposition is proved in [19].

Proposition 5. (1) If all the functions 𝑓^𝑁𝑗(⋅),𝑗∈𝑄_𝑁 are continuously differentiable, then 𝛾_{𝑞𝑁,𝑡𝑁}(⋅) is continuously differentiable and ∇_𝑥𝛾_{𝑞𝑁,𝑡𝑁}(𝑥)=^𝑞𝑁_𝑖=1𝜆^𝑖_{𝑞𝑁,𝑡𝑁}(𝑥)∇_𝑥𝑓^𝑁𝑖(𝑥)=^𝑘_𝑗=1𝜆^𝑖𝑗_{𝑞𝑁,𝑡𝑁}(𝑥)∇_𝑥𝑓^𝑁𝑖𝑗(𝑥), (17) where 𝜆^𝑖𝑗_{𝑞𝑁,𝑡𝑁}(𝑥)=1−3^𝑘−1_𝑙=1𝑐_𝑙ℎ_𝑙𝑥,𝑡_𝑁²3𝑐,𝑓𝑜𝑟𝑗=1,𝑗−1_𝑗−1ℎ_𝑗−1𝑥,𝑡_𝑁²−3^𝑘−1_𝑙=𝑗𝑐_𝑙ℎ_𝑙𝑥,𝑡_𝑁²3()𝑐,𝑓𝑜𝑟2≤𝑗<𝑘,𝑘−1_𝑘−1ℎ_𝑘−1𝑥,𝑡_𝑁²,𝑓𝑜𝑟𝑗=𝑘,0,𝑓𝑜𝑟𝑘<𝑗≤𝑞_𝑁, (18) and ℎ_𝑙(𝑥,𝑡_𝑁)=𝑙𝑓^{𝑁𝑖𝑙+1}(𝑥)−∑^𝑙_𝑟=1𝑓^𝑁𝑖𝑟(𝑥)+𝑡_𝑁.
(2) For any 𝑥∈𝑅^𝑛 and 𝑡_𝑁>0, 𝜆^𝑖𝑗_{𝑞𝑁,𝑡𝑁}(𝑥)∈[0,1), 𝑖_𝑗∈𝑄_𝑁 and ∑_{𝑖𝑗∈𝑄𝑁}𝜆^𝑖𝑗_{𝑞𝑁,𝑡𝑁}(𝑥)=1.
(3) If all the functions, 𝑓^𝑁𝑗(⋅),𝑗∈𝑄_𝑁, are twice continuously differentiable, then 𝛾_{𝑞𝑁,𝑡𝑁}(⋅) is twice continuously differentiable and ∇²_𝑥𝑥𝛾_{𝑞𝑁,𝑡𝑁}(𝑥)=^𝑘_𝑗=1𝜆^𝑖𝑗_{𝑞𝑁,𝑡𝑁}(𝑥)∇²_𝑥𝑥𝑓^𝑁𝑖𝑗(𝑥)+^𝑘_𝑗=1^𝑘_̆𝚥=1𝜆^{𝑖̆𝚥𝑗,}_{𝑞𝑁,𝑡𝑁}(𝑥)∇_𝑥𝑓^{𝑁𝑖̆𝚥}(𝑥)×∇_𝑥𝑓^𝑁𝑖𝑗(𝑥)^𝑇, (19) where 𝜆^𝑖̆𝚥1,_{𝑞𝑁,𝑡𝑁}(𝑥)=6^𝑘−1_𝑙=1𝑐_𝑙ℎ_𝑙𝑥,𝑡_𝑁̆̆𝑐𝑤ℎ𝑒𝑛𝚥=1,−6𝚥−1_̆𝚥−1ℎ_̆𝚥−1𝑥,𝑡_𝑁+6^𝑘−1_̆𝚥𝑙=𝑐_𝑙ℎ_𝑙𝑥,𝑡_𝑁̆()𝑐𝑤ℎ𝑒𝑛2≤𝚥<𝑘,−6𝑘−1_𝑘−1ℎ_𝑘−1𝑥,𝑡_𝑁̆𝜆𝑤ℎ𝑒𝑛𝚥=𝑘,_{𝑖̆𝚥𝑗,𝑞𝑁,𝑡𝑁}(𝑥)=−6𝑗−1𝑐_𝑗−1ℎ_𝑗−1𝑥,𝑡_𝑁+6^𝑘−1_𝑙=𝑗𝑐_𝑙ℎ_𝑙𝑥,𝑡_𝑁̆̆𝑐𝑤ℎ𝑒𝑛1≤𝚥<𝑗,−6𝚥−1_̆𝚥−1ℎ_̆𝚥−1𝑥,𝑡_𝑁+6^𝑘−1_̆𝚥𝑙=𝑐_𝑙ℎ_𝑙𝑥,𝑡_𝑁̆6̆𝑤ℎ𝑒𝑛𝑗<𝚥<𝑘,𝚥−1²𝑐_̆𝚥−1ℎ_̆𝚥−1𝑥,𝑡_𝑁+6^𝑘−1_̆𝚥𝑙=𝑐_𝑙ℎ_𝑙𝑥,𝑡_𝑁̆()𝑐𝑤ℎ𝑒𝑛𝚥=𝑗,−6𝑘−1_𝑘−1ℎ_𝑘−1𝑥,𝑡_𝑁̆𝜆𝑤ℎ𝑒𝑛𝚥=𝑘,𝑓𝑜𝑟2≤𝑗<𝑘,_{𝑖̆𝚥𝑘,𝑞𝑁,𝑡𝑁}(𝑥)=−6(𝑘−1)𝑐_𝑘−1ℎ_𝑘−1𝑥,𝑡_𝑁̆6()𝑤ℎ𝑒𝑛1≤𝚥<𝑘,𝑘−1²𝑐_𝑘−1ℎ_𝑘−1𝑥,𝑡_𝑁𝑤ℎ𝑒𝑛𝚥̆=𝑘. (20)

From Proposition 5, we obtain the following results.

Corollary 6. For any 𝑥∈𝑅^𝑛 and 𝑖_𝑗∈𝑄_𝑁, lim_𝑡𝑁→0𝜆^𝑖𝑗_{𝑞𝑁,𝑡𝑁}(𝑥)=1Ω(𝑥),𝑖𝑓𝑖_𝑗∈Ω(𝑥),0,𝑖𝑓𝑖_𝑗∉Ω(𝑥),lim_𝑡𝑁→0∇_𝑥𝛾_{𝑞𝑁,𝑡𝑁}(𝑥)=_{Ω(𝑥)𝑗∈}1Ω(𝑥∇)^𝑥𝑓^𝑗(𝑥). (21)

Proof. According to the definition of Δ_{𝑖1,…,𝑖𝑘}(𝑡_𝑁), we know that 𝑘→|Ω(𝑥)| as 𝑡_𝑁→0. From Proposition 5 (1), we know lim_𝑡𝑁→0𝜆^𝑖1_{𝑞𝑁,𝑡𝑁}(𝑥)=1−3^{|Ω(𝑥)|−1}_𝑙=1𝑐_𝑙𝑡²_𝑁=1−3^{|Ω(𝑥)|−1}_𝑙=113𝑙(𝑙+1)𝑡²_𝑁𝑡²_𝑁=1Ω(𝑥.) (22) If 𝑖_𝑗∈Ω(𝑥),(1<𝑗<𝑘), then lim_𝑡𝑁→0𝜆^𝑖𝑗_{𝑞𝑁,𝑡𝑁}(𝑥)=3𝑗−1𝑐_𝑗−1𝑡²_𝑁−3^{|Ω(𝑥)|−1}_𝑙=𝑗𝑐_𝑙𝑡²_𝑁1=3𝑗−13𝑗−1𝑗𝑡²_𝑁𝑡²_𝑁−3^{|Ω(𝑥)|−1}_𝑙=𝑗13𝑡²_𝑁1𝑙−1𝑡𝑙+1^2𝑁=1Ω(𝑥.) (23) If 𝑖_𝑘∈Ω(𝑥), then 𝑘=|Ω(𝑥)|. That is lim_𝑡𝑁→0𝜆^𝑖𝑘_{𝑞𝑁,𝑡𝑁}(𝑥)=3Ω(𝑥)−1𝑐_{|Ω(𝑥)|−1}𝑡²_𝑁=1Ω(𝑥.) (24) Then, lim_𝑡𝑁→0𝜆^𝑖𝑗_{𝑞𝑁,𝑡𝑁}(𝑥)=1/|Ω(𝑥)|,𝑖_𝑗∈Ω(𝑥).
If 𝑖_𝑗∉Ω(𝑥),(1<𝑗<𝑘), then lim_𝑡𝑁→0𝜆^𝑖𝑗_{𝑞𝑁,𝑡𝑁}(𝑥)=0.
It now follows from (21) and (17) that (21) holds. The proof is completed.

Next, let {𝑡_𝑁}^∞_𝑁=1 be an infinite sequence such that 𝑡_𝑁→0, as 𝑁→∞, and consider the sequence of approximating problems 𝑃_{𝑞𝑁,𝑡𝑁}min_{𝑥∈𝑅𝑛}𝛾_{𝑞𝑁,𝑡𝑁}(𝑥),𝑁=1,2,3,…, (25) with 𝛾_{𝑞𝑁,𝑡𝑁}(𝑥) defined as in (12).

Theorem 7. The problems (𝑃_{𝑞𝑁,𝑡𝑁}) epiconverge to the problem 𝑃.

Proof. Referring to Theorem  3.3.2 in [1], we see that to prove the theorem it is sufficient to show that if {𝑥_𝑁}^∞_𝑁=1 is a sequence in 𝑅^𝑛 converging to a point 𝑥, then 𝛾_{𝑞𝑁,𝑡𝑁}(𝑥)→𝜙(𝑥).
Thus, suppose that 𝑥_𝑁→𝑥 as 𝑁→∞. Then, 𝛾_{𝑞𝑁,𝑡𝑁}𝑥_𝑁(𝑥≤𝛾−𝜙)_{𝑞𝑁,𝑡𝑁}𝑥_𝑁−𝜙_𝑁𝑥_𝑁+𝜙_𝑁𝑥_𝑁−𝜙(𝑥). (26) Now, by Theorem 2, 𝜙_𝑁(𝑥_𝑁)−𝜙(𝑥)→0 as 𝑁→∞, and, since by assumption of 𝑡_𝑁→0, as 𝑁→∞, it follows from (14) that 𝛾_{𝑞𝑁,𝑡𝑁}(𝑥_𝑁)−𝜙_𝑁(𝑥_𝑁)→0 as 𝑁→∞, which completes our proof.

2. Spline Smoothing Newton Method and Its Convergence

We combine Algorithm  3.1 in [18] with a discretization precision test to produce an algorithm for solving the semi-infinite minimax problems 𝑃. The Hessian ∇²_𝑥𝑥𝛾_{𝑞𝑁,𝑡𝑁}(𝑥) of the smoothing spline function 𝛾_{𝑞𝑁,𝑡𝑁}(𝑥) can be modified by adding a multiple of the identity introduced in [20]; that is, 𝐵_{𝑞𝑁,𝑡𝑁}(𝑥)=𝜃(𝑥)𝐼+∇²_𝑥𝑥𝛾_{𝑞𝑁,𝑡𝑁}(𝑥), (27) where 𝜃(𝑥)=max{0,𝜗−𝑒(𝑥)} with 𝑒(𝑥) denoting the minimum eigenvalue of ∇²_𝑥𝑥𝛾_{𝑞𝑁,𝑡𝑁}(𝑥) and 𝜗>0.

Algorithm 8.
Data. Given 𝑥⁽⁰⁾∈𝑅^𝑛, a monotone increasing sequence of sets {𝑌_𝑁}^∞_𝑁=0, 𝑌_𝑁={𝑦₁,𝑦₂,…,𝑦_𝑞𝑁}, of cardinality 𝑞_𝑁, with 𝑞_𝑁→∞ as 𝑁→∞, satisfying Assumption 1, and defining the functions 𝑓^𝑁𝑗(𝑥)=𝜓(𝑥,𝑦_𝑁𝑗),𝑗=1,2,…,𝑞_𝑁, a sequence of monotone decreasing parameters {𝜋_𝑁}^∞_𝑁=0>0, such that 𝜋_𝑁→0 as 𝑁→∞, 𝜗>0, and 𝛿>0. Functions 𝜖_𝑎(𝑡),𝜖_𝑏(𝑡),𝜏(𝑡): (0,∞)→(0,∞), satisfying 𝜖_𝑏(𝑡)≥𝜖_𝑎(𝑡)>𝜏(𝑡), for all 𝑡>0, 𝑡̂≫1, 𝜅₁∈(0,1),𝜅₂≫1,0<𝜅₃≪1.

Parameter. Set 𝛼,𝛽∈(0,1).

Step 0. Set 𝑁=0.

Step 1. Set 𝑖=0,𝑡₀=𝜋_𝑁,𝑘=0,𝑠=1,𝑙=9,𝑥^𝑘,𝑖=𝑥⁽⁰⁾.

Step 2. Let 𝐼={𝑗∣max{𝑓^𝑁1(𝑥^𝑘,𝑖),…,𝑓^𝑁𝑞𝑁(𝑥^𝑘,𝑖)}−𝑓^𝑁𝑗(𝑥^𝑘,𝑖)<𝑡_𝑘}; let 𝑘 be the cardinality of 𝐼, and 𝐼={𝑖₁,𝑖₂,…,𝑖_𝑘}. Range {𝑓^𝑁𝑖𝑗(𝑥^𝑘,𝑖)}^𝑘_𝑗=1 according to 𝑓^𝑁𝑖1(𝑥^𝑘,𝑖)≥𝑓^𝑁𝑖2(𝑥^𝑘,𝑖)≥⋯≥𝑓^𝑁𝑖𝑘(𝑥^𝑘,𝑖).

If 𝑘=1, the cell is Δ_𝑖1(𝑡_𝑘).

Else if (𝑘−1)𝑓^𝑁𝑖𝑘(𝑥^(𝑘,𝑖))−∑^𝑘−1_𝑗=1𝑓^𝑁𝑖𝑗(𝑥^(𝑘,𝑖))+𝑡_𝑘≥0, for every 𝑘∈{𝑘,𝑘−1,…,2}, we have 𝑘∈𝐼⊆{𝑘,𝑘−1,…,2}. Let 𝑘 be the maximum element of 𝐼; then the cell is Δ_{𝑖1,…,𝑖𝑘}(𝑡_𝑘).

Step 3. Compute ∇_𝑥𝛾_{𝑞𝑁,𝑡𝑘}(𝑥^𝑘,𝑖). If ‖∇_𝑥𝛾_{𝑞𝑁,𝑡𝑘}(𝑥^𝑘,𝑖)‖²>𝜏(𝑡_𝑘), go to Step  4. Else go to Step  9.

Step 4. Compute 𝐵_{𝑞𝑁,𝑡𝑘}(𝑥^𝑘,𝑖) according to (27); then compute Cholesky factor 𝑅 such that 𝐵_{𝑞𝑁,𝑡𝑘}(𝑥^𝑘,𝑖)=𝑅𝑅^𝑇 and the reciprocal condition number 𝑐(𝑅) of 𝑅.

If 𝑐(𝑅)≥𝜅₁ and 𝑡_𝑘≥𝜅₃, go to Step  5.

Else if 𝑐(𝑅)≥𝜅₁ and the largest eigenvalue 𝜎_𝑡𝑘,max(𝑥^𝑘,𝑖) of 𝐵_{𝑞𝑁,𝑡𝑘}(𝑥^𝑘,𝑖) satisfies 𝜎_𝑡𝑘,max(𝑥^𝑘,𝑖)≤𝜅₂, go to Step  5.

Else go to Step  6.

Step 5. Compute the search direction ℎ_𝑘,𝑖=−𝐵_{𝑞𝑁,𝑡𝑘}𝑥^{𝑘,𝑖−1}∇_𝑥𝛾_{𝑞𝑁,𝑡𝑘}𝑥^𝑘,𝑖; (28) go to Step  7.

Step 6. Compute the search direction ℎ_𝑘,𝑖=−∇_𝑥𝛾_{𝑞𝑁,𝑡𝑘}𝑥^𝑘,𝑖. (29)

Step 7. Compute the step length 𝜆_𝑘,𝑖=𝛽^𝑙, where 𝑙≥0 is the smallest integer satisfying 𝛾_{𝑞𝑁,𝑡𝑘}𝑥^𝑘,𝑖+𝛽^𝑙ℎ_𝑘,𝑖−𝛾_{𝑞𝑁,𝑡𝑘}𝑥^𝑘,𝑖≤𝛼𝛽^𝑙∇_𝑥𝛾_{𝑞𝑁,𝑡𝑘}𝑥^𝑘,𝑖,ℎ_𝑘,𝑖. (30)

Step 8. Set 𝑥^𝑘,𝑖+1=𝑥^𝑘,𝑖+𝜆_𝑘,𝑖ℎ_𝑘,𝑖, 𝑖=𝑖+1. Go to Step  2.

Step 9. If 𝑠=1, compute 𝑡^∗ such that 𝜖_𝑎𝑡_𝑘≤∇_𝑥𝛾_{𝑞𝑁,𝑡∗}𝑥^𝑘,𝑖2≤𝜖_𝑏𝑡_𝑘; (31) go to Step  10.

Else set 𝑡_𝑘+1=1/𝑠(𝑘+2), 𝑘=𝑘+1, and 𝑖=0; go to Step  2.

Step 10. If 𝑡^∗≥𝑡̂, set 𝑡_𝑘+1=min{𝑡^∗,𝑡_𝑘/(𝑡_𝑘+1)}, 𝑘=𝑘+1, and 𝑖=0; go to Step  2. Else set 𝑠=max{2,((1/𝑡̂)+2)/(𝑘+1)}, 𝑡_𝑘+1=1/𝑠(𝑘+2), 𝑘=𝑘+1, and 𝑖=0; go to Step  2.

Step 11. If 12∇_𝑥𝛾_{𝑞𝑁,𝑡𝑘}(𝑥^𝑘,𝑖)²≤𝛿,_{𝑞𝑁𝑗=1}𝜆^𝑗_{𝑞𝑁,𝑡𝑘}𝑥^𝑘,𝑖𝜙_𝑁𝑥^𝑘,𝑖−𝑓^𝑁𝑗𝑥^𝑘,𝑖≤𝛿, (32) where the 𝜆^𝑗_{𝑞𝑁,𝑡𝑘}(𝑥^𝑘,𝑖) are defined by (17), set 𝜉_𝑁=𝑥^𝑘,𝑖, 𝑡_𝑁=𝑡_𝑘, replace 𝑁 by 𝑁+1, 𝛿 by 𝛿/2, and go to Step  1.

Else go to Step  1.

Theorem 9. Suppose that {𝜉_𝑁}^∞_𝑁=0 is a sequence constructed by Algorithm 8. Then, any accumulation point 𝜉̂ of this sequence satisfies 𝜃(𝜉̂)=0.

Proof. First note that it follows from Corollary 6 that condition (32) will be eventually satisfied, since Algorithm 8 keeps abating 𝑡_𝑘. Next, note that −^𝑞𝑁_𝑗=1𝜆^𝑗_{𝑞𝑁,𝑡𝑁}𝜙_𝑁𝜉_𝑁−𝑓^𝑁𝑗𝜉_𝑁−12∇_𝑥𝛾_{𝑞𝑁,𝑡𝑁}(𝜉_𝑁)²≤𝜌_𝑁𝜉_𝑁≤0. (33) Since by construction −^𝑞𝑁_𝑗=1𝜆^𝑗_{𝑞𝑁,𝑡𝑁}𝜙_𝑁𝜉_𝑁−𝑓^𝑁𝑗𝜉_𝑁−12∇_𝑥𝛾_{𝑞𝑁,𝑡𝑁}𝜉_𝑁²⟶0,as𝑁⟶∞, (34) and 𝜌_𝑁(𝜉_𝑁)→^𝑘𝜌(𝜉̂) on any infinite subsequence {𝜉_𝑁}_𝑁∈𝐾 that converges to 𝜉̂, the desired result follows.

3. Numerical Experiment

We have implemented Algorithm 8 using Matlab. In order to show the efficiency of the algorithm, we also have implemented algorithm in [16] (denote PWY) using similar procedures. Algorithm PWY was proposed by Polak et al. in [16], which has been introduced in Section 1.

The test results were obtained by running Matlab R2011a on a desktop with Windows XP Professional operation system, Intel(R) Core (TM) i3-370 2.40 GHz processor, and 2.92 GB of memory.

In Algorithm 8, parameters are chosen as follows: 𝛼=0.8, 𝛽=0.77, 𝑡̂=10⁵ln𝑞_𝑁, 𝜅₁=10⁻⁷, 𝜅₂=10³⁰, 𝜅₃=1000𝑡̂, 𝜏(𝑡)=min{0.1,1000𝑡/(𝑁+1)²}, (𝜖_𝑎,𝜖_𝑏)=(0.01,0.2), 𝜗=0.1, 𝛿=1, 𝑞_𝑁=100⋅2^𝑁+1, and 𝜋_𝑁=1/(𝑁²+(1/𝑝_𝑁)). In the PWY algorithm, parameters are chosen as follows: 𝜀=0.1, 𝜏(𝑝)=min{0.1,1000/((𝑁+1)²𝑝)}. The results are listed in Tables 1, 2, 3, 4, and 5. 𝑥^∗ denotes the final approximate solution point and 𝜙^∗ is the value of the objective function of discretized problems 𝑃_𝑌𝑁 at 𝑥^∗. 𝑚 is the maximum number of discrete points. Time is the CPU time in seconds.

Table 1 

Test results for Example 1.

Table 2 

Test results for Example 2.

Table 3 

Test results for Example 3.

Table 4 

Test results for Example 4.

Table 5 

Test results for Example 5.

Example 1 (see [21]). Let 𝑥=(𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅,𝑥₆)∈𝑅⁶𝜙𝑥,𝑦=𝑥²₁exp−𝑥₂𝑦cos²𝑥₃𝑦+𝑥₄−cos𝑦+𝑥²₂𝑥²₃exp−𝑥₁𝑦sin²𝑥₂𝑦+exp1−𝑥₆²𝑦+x²₅,𝑌=[0,10]. (35)

Example 2 (see [21]). Let 𝑥=(𝑥₁,𝑥₂,𝑥₃,𝑥₄)∈𝑅⁴𝜙𝑥,𝑦=𝑥₁+𝑥²₁+cos𝑦+4×lnsin𝑦+expcos𝑥₁+𝑥²₂+−𝑥exp−𝑦³+𝑦𝑥₄1+𝑥²₃²,𝑌=[0,10]. (36)

Example 3. Let 𝑥=(𝑥₁,𝑥₂,𝑥₃,𝑥₄)∈𝑅⁴𝜙𝑥,𝑦=𝑥²₂𝑥²₃exp−𝑥₁𝑦sin²𝑥₂𝑦−𝑥₁cos𝑦−𝑥₄𝑦[].sin+5,𝑌=0,2𝜋 (37)

Example 4. Let 𝑥=(𝑥₁,𝑥₂)∈𝑅²𝜙𝑥,𝑦=𝑥²₁+2𝑥₁𝑦²+𝑒^𝑥1+𝑥2−𝑒^𝑦,𝑌=[0,10]. (38)

Example 5. Let 𝑥=(𝑥₁,𝑥₂,𝑥₃,𝑥₄)∈𝑅⁴𝜙=𝑦𝑥,𝑦¹−𝑥₁²𝑥²₃+𝑦₂−𝑥₂²𝑥²₄,[]×[].−1𝑌=0,10,1 (39)

4. Conclusion

We have developed a spline smoothing Newton method for the solution of semi-infinite minimax problems using smooth cubic spline and discretization strategy. At each iteration, only few components in the max function are computed; hence, the computation cost is greatly reduced. For semi-infinite minimax problems with complicated component functions, numerical tests show that the new method is very efficient.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11171051, 91230103) and the Fundamental Research Funds for the Central Universities (DC13010214) and the General Project of the Education Department of Liaoning Province (0908-330006).