Abstract
This paper proposes an implementable SAA (sample average approximation) nonlinear Lagrange algorithm for the constrained minimax stochastic optimization problem based on the sample average approximation method. A computable nonlinear Lagrange function with sample average approximation functions of original functions is minimized and the Lagrange multiplier is updated based on the sample average approximation functions of original functions in the algorithm. And it is shown that the solution sequences obtained by the novel algorithm for solving subproblem converge to their true counterparts with probability one as the sample size approximates infinity under some moderate assumptions. Finally, numerical experiments are carried out for solving some typical test problems and the obtained numerical results preliminarily demonstrate that the proposed algorithm is promising.
1. Introduction
Minimax stochastic optimization is a kind of important problem in stochastic optimization. Minimax stochastic optimization has drawn much attention in recent years, which has been widely applied in subjects such as inventory theory, finance optimization, control science, and engineering field (see [1–7]).
This paper considers the constrained minimax stochastic optimization problems as follows:where , is a random vector that is defined on the probability space , denotes mathematical expectation with respect to the distribution of , and , and are well-defined. Since the objective function is not differentiable, the efficient smooth optimization methods cannot be used to solve the problem (1) directly. Moreover, either distribution of random vector is unknown or it is too complex to compute the multidimensional integral, so the exact numerical evaluation of the expected value functions in problem (1) is very difficult, which results in that problem (1) cannot be solved directly by the traditional deterministic optimization methods even though problem (1) is smooth.
On the one hand, as one of the famous smoothing techniques aiming to overcome the nonsmoothness of (see [1, 8–16]), the nonlinear Lagrange method has many interesting merits, such as no restrictions on the feasibility of variables , improvement on the convergence rate and the numerical robustness compared with penalty method by introducing the Lagrangian multipliers as the main driving force. On the other hand, the sample average approximation method is one of the well-behaved approaches for solving stochastic programming problems, the basic idea of which is to generate an independent and identically distributed (i.i.d.) sample of the random variable with sample size by Monte Carlo sampling method and approximate the involved expected value functions in problem (1) by their corresponding sample average functions. The SAA method has drawn much attention from many authors, see the comprehensive work by Shapiro [17] and the other works in [18–30].
Motivated by the effectiveness of the nonlinear Lagrange method, this paper presents a nonlinear Lagrange function for problem (1) based on the work in [13] as follows:where , , , , , , , , is Lagrange multiplier, , is a penalty parameter, and is an estimate of the objective function . has good properties and the corresponding nonlinear Lagrangian algorithm is recalled below (see [13]).
Algorithm 1.
Step 1. Choose , where , , , , , and being small enough and set .
Step 2. Solveand obtain the optimal solution .
Step 3. If , then stop; otherwise go to Step 4.
Step 4. Update the Lagrange multiplier , and by Step 5. Set and return to Step 2.
In view of the difficulty in the numerical computation of the expected value function in Algorithm 1 and the inspiration from the SAA method, this paper will propose an implementable SAA nonlinear Lagrange algorithm in Section 3. And under some suitable assumptions, the convergence of the proposed algorithm will be analyzed in Section 3. In Section 2, some useful preliminaries will be presented. Furthermore, the numerical results for some typical test problems are reported to verify the feasibility and effectiveness of the proposed algorithm in Section 4. In the last section, the conclusion is drawn.
2. Preliminaries
This section serves as a preparation for the theoretical analysis in the subsequent section. Firstly, this section provides some assumptions on problem (1), and then recalls some related definitions and conclusions.
The Lagrange function for problem (1) is defined by . Let denote the Karush-Kuhn-Tucker (KKT) solution of problem (1) and (see [13]). Let be small enough. Define , , , , and . Define . Some assumptions on problem (1) are made as follows: (a)For any , , , are twice continuously differentiable with respect to on , and the function values are finite, where , , and (b)There exist nonnegative measurable functions such that being finite and for every the following inequalities are true with probability one: (c)The random sample is independent and identically distributed, and obeys the law of large numbers.(d)For ease of presentation, assume (e)KKT condition holds. That is, (f)Strict complementarity condition holds, i.e., for and for (g)The vectors of are linearly independent.(h)There exists a constant such that, for all in satisfying , , , , and , , it holds that
The following definition (see [17]) is recalled.
Definition 2. For nonempty sets and in , we denote by the distance from to , and by the deviation of the set from the set .
To present the basic lemma, we now consider the following stochastic optimization problem:where is a nonempty and compact subset on , is a random vector on , . For any , is finite and continuous for all . The sample average approximation problem of (9) can be expressed aswhere , and are independent sample observations and obey the law of large numbers. Let and denote the optimal value and the optimal solution set of problem (9), and indicate the optimal value and the optimal solution set of problem (10). Then the resulted essential conclusion is obtained in Lemma 3.
Lemma 3 (see [17]). Suppose that there exists a nonnegative measurable function independent of for such that with probability one. Then the following conclusions are true: (i) is continuous and finite on ;(ii) converges to with probability one uniformly on as ;(iii) converges to with probability one as ;(iv) converges to 0 with probability one as .
3. The SAA Nonlinear Lagrange Algorithm and Its Convergence
This section presents an implementable SAA nonlinear Lagrange algorithm based on the SAA nonlinear Lagrange function of nonlinear Lagrange function (2) and then analyzes its convergence by means of the preliminaries in Section 2.
Firstly, we construct a SAA nonlinear Lagrange function of nonlinear Lagrange function (2) below:where denotes , , , , and is a random sample.
Based on the SAA nonlinear Lagrange function (11) and Algorithm 1, an implementable SAA nonlinear Lagrange algorithm is presented as follows.
Algorithm 4.
Step 1. Choose , where , being small enough, , , , , and is large enough. Set .
Step 2. Solveand obtain the optimal solution .
Step 3. If , then stop; otherwise go to.
Step 4. Update the Lagrange multiplier , and by
Step 5. Set and return to Step 2.
Next, we study the convergence of Algorithm 4 on based on the assumptions (a)-(h) and Lemma 3 in Section 2. Let and denote the optimal value and optimal solution set of problem (12), and denote the optimal value and optimal solution set of problem (3).
Theorem 5. If assumptions (a)-(c) hold and for , converges to with probability one, converges to with probability one, converges to with probability one, and converges to with probability one as , then the following statements are true. (i) converges to with probability one uniformly as on ;(ii) converges to with probability one and converges to 0 with probability one as .
Proof. (i) Define , , and , where and is defined as in (2). And setAt first, we are to prove that converges to with probability one uniformly as , for which we need to prove that as , , converges to with probability one uniformly, converges to with probability one uniformly, and converges to with probability one uniformly on , respectively. The proof for it is divided into the following three parts.
(A) First we shall prove that as , converges to with probability one uniformly on .
According to the definition of , we haveOne has that and are continuous on with respect to from the assumption (a) and Lemma 3, so there exists a closed interval such that , and for . Since and are bounded, there exist , , such that and . Thus, we have that is bounded on with respect to . Let . Then is continuous in with respect to . It follows that is continuous uniformly in from the property of continuous function. That is, for any , there exists , for ; if , it holds thatNote that converges to with probability one uniformly on from Lemma 3 and converges to with probability one; hence for , there exists , when , for any , we haveIt follows from (16) and (17) that for any , there exists , when ; for any , it holds thatThus, in view of (15), , and the condition that converges to with probability one, we obtain that converges to with probability one uniformly on as .
(B) Next, we prove that converges to with probability one uniformly on as .
From the definition of , we haveSince and are continuous on with respect to from the assumption (a) and Lemma 3, there exits a closed interval such that and for , which means that is bounded on with respect to . From the proof process of (A), we get that, for any , there exists a positive integer , when , for , it holds that Therefore, combined with (19), it follows from converging to with probability one as and that converges to uniformly with probability one on as .
(C) Now we prove that converges to with probability one uniformly on as .
From the definition of , we haveSince and are continuous on with respect to according to the assumption (a) and Lemma 3, there exists a closed interval such that and for ; i.e., is bounded. Then from it follows that is bounded on . From Lemma 3, it is true that converges to with probability one uniformly on . That is, for any , there exists a positive integer , when , for , the following inequality holds with probability one: Moreover, considering (21) and the fact that converges to with probability one as , we have that converges to with probability one uniformly on as .
Thus, from the above analyses of (A), (B) and (C), we draw the conclusion that converges to with probability one uniformly on as . Furthermore, in view of the fact that is continuous with respect to on , it can be proven that converges to uniformly with probability one as for , which implies that the conclusion (i) is true.
(ii) From the conclusion (i) and Lemma 3, we can prove that the conclusion (ii) is true.
Theorem 6. If assumptions (a)-(c) hold and letting , , , and , then for any , the following statements hold: (i)As , converges to with probability one for , converges to with probability one for , converges to with probability one for , and converges to with probability one;(ii) converges to with probability one uniformly on as ;(iii) converges to with probability one, and converges to 0 with probability one as .
Proof. (i) We use the mathematical induction method to show that the statement (i) is true below. For , we know that , , and , which means that the conclusion (i) is true for . Next we prove that the conclusion (i) is true for .
For , from , one has that For any , we haveFrom Lemma 3, we know that converges to with probability one as , which implies that the first term on the right side of (24) converges to 0 as . And we know that is continuous and bounded on from Lemma 3, and converges to with probability one as from Theorem 5, so the second part on the right side of (24) converges to 0 as . That is, converges to with probability one as . And we know is continuous with respect to on , hence converges to with probability one as . Furthermore, we can prove that converges to with probability one as for .
For , we have For any , , so we have Similarly, we can prove that converges to with probability one as for from Lemma 3 and Theorem 5.
For , since , it holds that Hence for any , we have We can also prove that converges to with probability one as for from Lemma 3 and Theorem 5.
For , it holds that Similar to the above proof process, we can draw the conclusion that converges to with probability one as .
That is, we have proven that the conclusion (i) is true for . Now suppose that the conclusion (i) is true for , where is an integer (). Next, we prove that the conclusion (i) is true for .
Since the conclusion (i) is true for , we have that converges to with probability one as from Theorem 5. For , we have For any , one hasIn view of the proof process in Theorem 5, it follows that converges to with probability one as . Moreover, note that the special forms of and , we can verify that converges to with probability one as for .
Similarly, we can prove that converges to with probability one as for , converges to with probability one as for , and converges to with probability one as from Lemma 3 and Theorem 5. That is, the conclusion (i) is true for . By mathematical induction method, hence we know that conclusion (i) is true for any .
(ii) Considering conclusion (i) and Theorem 5, we know that conclusion (ii) is true.
(iv) In view of conclusion (ii) and Lemma 3, we have that conclusion (iii) is true.
Thus, the proof of Theorem 6 is completed.
Up until now, we have established the relationship between the optimal solution of problem (3) and the optimal solution of problem (12), and the convergence of the SAA Lagrange multiplier sequence in Algorithm 4 with probability one as . Next we are to prove that the optimal solution sequence and the SAA Lagrange multiplier sequence obtained by Algorithm 4 converge to the optimal solution and the corresponding Lagrange multiplier of problem (1) with probability one as under the assumptions (a)-(h).
Theorem 7. If assumptions (a)-(h) hold, and let , , , , then there exist and such that for any , converges to with probability one, converges to with probability one, converges to with probability one, and converges to with probability one, respectively, when and .
Proof. Based on the property of norm, one hasIf assumptions (d)-(h) are satisfied, then it follows from Theorem 3.1 of [13] that there exist and such that as , , , , , and for any .
If assumptions (a)-(c) are satisfied, and , , and , it follows from Theorem 6 that converges to with probability one, converges to with probability one, converges to with probability one, converges to with probability one, and converges to with probability one as .
Thus, combined with (32)-(35) and the above analysis, it has been proven that Theorem 7 is true under assumptions (a)-(h).
Remark 8. Theorem 7 shows that Algorithm 4 is locally convergent under assumptions (a)-(h). That is, when the initial multiplier are close to the optimal multiplier , is close to , and is less than a threshold, the solution sequence obtained by Algorithm 4 locally converges to the optimal solution of original problem (1) with probability one as , converges to with probability one as , and converges to with probability one as under assumptions (a)-(h).
4. Numerical Results
In this section, the numerical results for eight test examples by using Algorithm 4 are presented. These test examples are compiled based on the deterministic optimization problems in [13] by considering random variable . The numerical experiments are implemented in Matlab2014 on the same computer whose CPU basic parameters are Intel CORE(TM) [email protected] and memory 8GB.
In the experiments, the random variable is set to be uniformly distributed on and the random sample with sample size is generated by random number generator in Matlab2014. We choose , , , , and to make comparison for each test example. The initial values of , , and are set as , , and for each example, and is chosen small enough and determined by the scale of test problem; Unconstrained minimization problem in Step 2 of Algorithm 4 is solved by BFGS quasi-Newton method combined with Wolf nonexact linear search rule. The stopping precision in Step 3 is , and the termination condition is
The obtained numerical results are reported in Tables 1-8, in which represents sample size; represents the value of ; represents the number of iterations, i.e., the numbers of the Lagrange multipliers being updated; represents the gap between the solution sequence obtained by Algorithm 4 and the optimal solution of the corresponding test problem; and represents the gap between the approximate value of objective function obtained by Algorithm 4 and the optimal value of the corresponding test problem, respectively.
Example 1. In problem (1), and are defined as follows: where the optimal solution and the optimal value (see [13]) are The numerical results for this example obtained by Algorithm 4 are shown in Table 1.
Example 2. In problem (1), , , and are defined as follows: where the optimal solution and the optimal value (see [13]) are The numerical results for this example obtained by Algorithm 4 are shown in Table 2.
Example 3. In problem (1), and are defined as follows: where the optimal solution and the optimal value (see [13]) are The numerical results for this example obtained by Algorithm 4 are shown in Table 3.
Example 4. In problem (1), and are defined as follows: where the optimal solution and the optimal value (see [13]) are The numerical results for this example obtained by Algorithm 4 are shown in Table 4.
Example 5. In problem (1), , , and are defined as follows: where the optimal solution and the optimal value (see [13]) are The numerical results for this example obtained by Algorithm 4 are shown in Table 5.
Example 6. In problem (1), , and are defined as follows: where the optimal solution and the optimal value (see [13]) are The numerical results for this example obtained by Algorithm 4 are shown in Table 6.
Example 7. In problem (1), , , and are defined as follows: where the optimal solution and the optimal value (see [13]) areThe numerical results for this example obtained by Algorithm 4 are shown in Table 7.
Example 8. In problem (1), , , and are defined as follows: where the optimal solution and the optimal value (see [13]) are The numerical results for this example obtained by Algorithm 4 are shown in Table 8.
From the numerical results in Tables 1-8, the following remarks are made.
Remark 9. The preliminary numerical results show that Algorithm 4 is feasible and promising.
Remark 10. Compared with the numerical results for the same test example with the different sample size , the numerical results in Tables 1-8 show that the precisions of the optimal solution and the optimal value by Algorithm 4 become higher as the sample size is chosen larger, which coincides with the convergence result of Algorithm 4 in Section 3.
5. Conclusions
An implementable SAA nonlinear Lagrange algorithm for solving constrained minimax stochastic optimization problems is presented by this paper. And the convergence theory of the proposed algorithm is established under some assumptions, in which the KKT solution sequence obtained by the algorithm is demonstrated to converge to the optimal KKT solution of the original problem with probability one as the sample size approaches to infinity. Furthermore, numerical experiments are implemented by using the proposed SAA nonlinear Lagrange algorithm for solving eight typical test examples, and the results of numerical experiment verify the convergence theory and indicate that the new algorithm is promising. Moreover, the numerical experiments for obtaining the solutions with higher precision and solving large scale problems deserve our future attention. And applying this proposed algorithm to solve some practical problems is also interesting.
Data Availability
The [numerical examples and results] data used to support the findings of this study are included within the article, which are given in Section 4 of the manuscript.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research is supported by the National Natural Science Foundation of China under project (no. 11201357) and the Fundamental Research Funds for the Central Universities under project (no. 2018IB016).