王鑫 彭定濤 周倩

摘 要 研究一類帶有閉凸集約束的稀疏約束非線性規(guī)劃問題，這類問題在變量選擇、模式識別、投資組合等領(lǐng)域具有廣泛的應(yīng)用.首先引進(jìn)了限制性Slater約束規(guī)格的概念，證明了該約束規(guī)格強(qiáng)于限制性M-F約束規(guī)格，然后在此約束規(guī)格成立的條件下，分析了其局部最優(yōu)解成立的充分和必要條件.最后，對約束集合的兩種具體形式，指出限制性Slater約束規(guī)格必滿足，并給出了一階必要性條件的具體表達(dá)形式.

關(guān)鍵詞 稀疏約束非線性規(guī)劃;限制性約束規(guī)格;最優(yōu)性條件

中圖分類號 O224 ?文獻(xiàn)標(biāo)識碼 A

Abstract A class of sparse nonlinear programming was studied， whose feasible set is the intersection of a closed convex set and a sparse set. This model is a typical sparse optimization problem which has wide applications in variable selection， pattern recognition， portfolio management and other fields. We defined ?the restricted Slater constraint qualification for this sparse nonlinear programming and proved ?that this restricted Slater constraint qualification is stronger than the restricted M-F constraint qualification. Under this restricted Slater constraint qualification， we analyzed ?the necessary or sufficient optimality conditions for the local solutions. Finally， we provided ?the specific expressions of the first-order necessary optimality condition for the model with two specific constraint sets.

Key words sparse constraint nonlinear programming; restricted constraint qualification; optimality condition

5 總 結(jié)

本文對一類帶有閉凸集約束的稀疏約束非線性規(guī)劃問題引進(jìn)了限制性Slater約束規(guī)格的概念，分析表明該約束規(guī)格強(qiáng)于限制性M-F約束規(guī)格且更容易驗(yàn)證，此約束規(guī)格可保證局部最優(yōu)解是M-KKT點(diǎn)、C-KKT點(diǎn)和B-穩(wěn)定點(diǎn).最后，對約束集合的兩種具體形式，指出限制性Slater約束規(guī)格必滿足，并給出了一階必要性條件的具體表達(dá)形式.本文的結(jié)果對于設(shè)計和分析有效算法提供了理論基礎(chǔ).

參考文獻(xiàn)

[1]王宜舉， 修乃華： 非線性最優(yōu)化理論與方法[M].北京：科學(xué)出版社，2012.

[2]BECK A， ELDAR ?Y C. Sparsity constrained nonlinear optimization：optimality conditions and algorithms[J]. SIAM Journal on Optimization， 2012， 23（3）：1480-1509.

[3]BECK A， HALLAK ?N. On the minimization over sparse symmetric sets：projections， optimality conditions， and algorithms[J]. Mathematics of Operations Research， 2016， 41（1）：196-223.

[4]PAN ?L L， XIU ?N H， FAN J. Optimality conditions for sparse nonlinear programming[J]. Science China， 2017， 60（5）：1-18.

[5]PAN ?L L， XIU ?N H， ZHOU ?S L. On solutions of sparsity constrained optimization[J]. Journal of the Operations Research Society of China， 2015， 3（4）：421-439.

[6]CERVINKA M， KANZOW C， SCHWARTZ ?A. Constraint qualifications and optimality conditions for optimization problems with cardinality constraints[J]. Mathematical Programming， 2016， 160（1/2）：353-377.

[7]ROCKAFELLAR ?R T， WETS ?R J. Variational analysis[M]. Berlin：Springer-verlag，1998.