Journal of Control Science and Engineering

Journal of Control Science and Engineering / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 127208 | https://doi.org/10.1155/2011/127208

S. K. Padhan, C. Nahak, "Higher-Order Generalized Invexity in Control Problems", Journal of Control Science and Engineering, vol. 2011, Article ID 127208, 9 pages, 2011. https://doi.org/10.1155/2011/127208

Higher-Order Generalized Invexity in Control Problems

Academic Editor: Onur Toker
Received13 Dec 2010
Accepted30 Mar 2011
Published26 May 2011

Abstract

We introduce a higher-order duality (Mangasarian type and Mond-Weir type) for the control problem. Under the higher-order generalized invexity assumptions on the functions that compose the primal problems, higher-order duality results (weak duality, strong duality, and converse duality) are derived for these pair of problems. Also, we establish few examples in support of our investigation.

1. Introduction

We consider the control problem๎€œ(CP)min๐‘๐‘Ž๐‘“(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))๐‘‘๐‘ก,(1)subjectto๐‘”(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))โ‰ค0,(2)๐บ(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))=0,(3)๐‘ฅ(๐‘Ž)=๐›พ1,๐‘ฅ(๐‘)=๐›พ2;๐‘ข(๐‘Ž)=๐›ฟ1,๐‘ข(๐‘)=๐›ฟ2,(4) where ๐‘“, ๐‘”, and ๐บ are twice continuously differentiable functions from ๐ผร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘š into โ„, โ„๐‘Ÿ, and โ„๐‘ , respectively, ๐ผ=[๐‘Ž,๐‘]โŠ†โ„.

Mangasarian [1] formulated a class of higher-order dual problems for a nonlinear programming problems involving twice continuously differentiable functions. He did not prove the weak duality and hence gave a limited strong duality theorem. Mond and Zhang [2] introduced invexity type conditions under which duality holds between Mangasarian [1] primal problem and various higher-order dual programming problems.

One practical advantage of higher-order duality is that it provides tighter bounds for the value of the objective function of the primal problem when approximations are used, because there are more parameters involved. Higher-order duality in nonlinear programming has been studied by several researchers like Mond and Zhang [2], Chen [3], Mishra and Rueda [4], and Yang et al. [5]. Recently, Gulati and Gupta [6] studied the higher-order symmetric duality over arbitrary cones for Wolfe and Mond-Weir type models. Obtained duality results for various higher-order dual problems under higher-order and type higher-order duality to higher-order type I functions.

Bhatia and Kumar [7] have studied the multiobjective control problems under ๐œŒ-pseudoinvexity, ๐œŒ-strictly pseudoinvexity, ๐œŒ-quasi-invexity, and ๐œŒ-strictly quasi-invexity assumptions. Nahak and Nanda [8] have studied the efficiency and duality for multiobjective control problems under (๐นโˆ’๐œŒ) convexity. Again Nahak and Nanda [9] proposed a sufficient condition for solutions and duality for the multiobjective variational control problems under ๐‘‰-invexity. Recently, Padhan and Nahak [10] considered a class of constrained nonlinear control primal problem and formulated the second-order dual. He also gave some duality results (weak duality, strong duality, and converse duality) under generalized invexity assumptions. But in our knowledge, no one has talked about higher-order duality for the control problem. In this paper, we study both Mangasarian and Mond-Weir type higher-order duality of the control primal problem (CP). We give more general type conditions that is higher-order generalized invexity under which duality holds between (CP) and (MHCD), and (CP) and (MWHVD). Our approach is similar to that of Mangasarian [1]. Again, we discuss many counterexamples to justify our work.

2. Notations and Preliminaries

Let ๐‘†(๐ผ,โ„๐‘›) denote the space of piecewise smooth functions ๐‘ฅ with norm โ€–๐‘ฅโ€–=โ€–๐‘ฅโ€–โˆž+โ€–๐ท๐‘ฅโ€–โˆž, where the differentiation operator ๐ท is given by๎€œ๐‘ข=๐ท๐‘ฅโŸบ๐‘ฅ(๐‘ก)=๐œ…+๐‘ก๐‘Ž๐‘ข(๐‘ )๐‘‘๐‘ ,(5) where ๐œ… is a given boundary value; thus, ๐‘‘/๐‘‘๐‘ก=๐ท except at discontinuities.

The higher-order generalized invexity functions are defined as follows.

Definition 2.1. The scalar functional โˆซ๐น(๐‘ฅ,๐‘ข)=๐‘๐‘Ž๐‘“(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))๐‘‘๐‘ก is said to be higher-order ๐œŒโˆ’(๐œ‚,๐œ‰,๐œƒ)-invex in ๐‘ฅ, ฬ‡๐‘ฅ, and ๐‘ข if there exist ๐œ‚,๐œƒโˆถ๐ผร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘šร—โ„๐‘šโ†’โ„๐‘›,๐œ‰โˆถ๐ผร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘šร—โ„๐‘šโ†’โ„๐‘š, โ„Žโˆถ๐ผร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘šร—โ„๐‘›โ†’โ„,โ„Ž1โˆถ๐ผร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘šร—โ„๐‘šโ†’โ„ and ๐œŒโˆˆโ„, such that โ‰ฅ๎€œ๐น(๐‘ฅ,๐‘ข)โˆ’๐น(๐‘ฆ,๐‘ฃ)๐‘๐‘Ž๎€บ๐œ‚(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ข(๐‘ก),๐‘ฃ(๐‘ก))๐‘‡โˆ‡๐‘โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+๐œ‰(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ข(๐‘ก),๐‘ฃ(๐‘ก))๐‘‡โˆ‡๐‘žโ„Ž1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)+โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+โ„Ž1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)โˆ’๐‘๐‘‡โˆ‡๐‘โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)โˆ’๐‘ž๐‘‡โˆ‡๐‘žโ„Ž1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)+๐œŒโ€–๐œƒ(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ข(๐‘ก),๐‘ฃ(๐‘ก))โ€–2๎€ป๐‘‘๐‘ก.(6)

Definition 2.2. The scalar functional โˆซ๐น(๐‘ฅ,๐‘ข)=๐‘๐‘Ž๐‘“(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))๐‘‘๐‘ก is said to be higher-order ๐œŒโˆ’(๐œ‚,๐œ‰,๐œƒ)-pseudoinvex in ๐‘ฅ, ฬ‡๐‘ฅ, and ๐‘ข if there exist ๐œ‚,๐œƒโˆถ๐ผร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘šร—โ„๐‘šโ†’โ„๐‘›,๐œ‰โˆถ๐ผร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘šร—โ„๐‘šโ†’โ„๐‘š, โ„Žโˆถ๐ผร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘šร—โ„๐‘›โ†’โ„,โ„Ž1โˆถ๐ผร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘šร—โ„๐‘šโ†’โ„ and ๐œŒโˆˆโ„, such that ๎€œ๐‘๐‘Ž๎€บ๐œ‚(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ข(๐‘ก),๐‘ฃ(๐‘ก))๐‘‡โˆ‡๐‘โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+๐œ‰(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ข(๐‘ก),๐‘ฃ(๐‘ก))๐‘‡โˆ‡๐‘žโ„Ž1โ€–(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)+๐œŒโ€–๐œƒ(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ข(๐‘ก),๐‘ฃ(๐‘ก))2๎€ปโˆ’๎€œ๐‘‘๐‘กโ‰ฅ0โŸน๐น(๐‘ฅ,๐‘ข)โˆ’๐น(๐‘ฆ,๐‘ฃ)๐‘๐‘Ž๎€บโ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+โ„Ž1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)โˆ’๐‘๐‘‡โˆ‡๐‘โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)โˆ’๐‘ž๐‘‡โˆ‡๐‘žโ„Ž1๎€ป(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)๐‘‘๐‘กโ‰ฅ0.(7)

Definition 2.3. The scalar functional โˆซ๐น(๐‘ฅ,๐‘ข)=๐‘๐‘Ž๐‘“(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))๐‘‘๐‘ก is said to be higher-order ๐œŒโˆ’(๐œ‚,๐œ‰,๐œƒ)-quasi-invex in ๐‘ฅ, ฬ‡๐‘ฅ and ๐‘ข, if there exist ๐œ‚,๐œƒโˆถ๐ผร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘šร—โ„๐‘šโ†’โ„๐‘›, ๐œ‰โˆถ๐ผร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘šร—โ„๐‘šโ†’โ„๐‘š, โ„Žโˆถ๐ผร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘šร—โ„๐‘›โ†’โ„, โ„Ž1โˆถ๐ผร—โ„๐‘›ร—โ„๐‘›ร—โ„๐‘šร—โ„๐‘šโ†’โ„ and ๐œŒโˆˆโ„, such that โˆ’๎€œ๐น(๐‘ฅ,๐‘ข)โˆ’๐น(๐‘ฆ,๐‘ฃ)๐‘๐‘Ž๎€บโ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+โ„Ž1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)โˆ’๐‘๐‘‡โˆ‡๐‘โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)โˆ’๐‘ž๐‘‡โˆ‡๐‘žโ„Ž1๎€ปโŸน๎€œ(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)๐‘‘๐‘กโ‰ค0๐‘๐‘Ž๎€บ๐œ‚(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ข(๐‘ก),๐‘ฃ(๐‘ก))๐‘‡โˆ‡๐‘โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+๐œ‰(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ข(๐‘ก),๐‘ฃ(๐‘ก))๐‘‡โˆ‡๐‘žโ„Ž1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)+๐œŒโ€–๐œƒ(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ข(๐‘ก),๐‘ฃ(๐‘ก))โ€–2๎€ป๐‘‘๐‘กโ‰ค0.(8)

Remark 2.4. If โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)=๐‘๐‘‡๎‚ƒ๐‘“๐‘ฅ๐‘‘(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))โˆ’๐‘“๐‘‘๐‘กฬ‡๐‘ฅ๎‚„,โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)=๐‘ž๐‘‡๐‘“๐‘ข(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก)),(9) with ๐œ‚=0 at ๐‘ก=๐‘Ž and ๐‘ก=๐‘, then the above definitions becomes the definitions of invexity defined by Padhan and Nahak [10].

3. Mangasarian Type Higher-Order Duality

In this section, we propose the following Mangasarian type higher-order dual (MHCD) to (CP): ๎€œ(MHCD)max๐‘๐‘Ž๎€บ๐‘“(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))+๐›ผ(๐‘ก)๐‘‡๐‘”(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))+๐›ฝ(๐‘ก)๐‘‡๐บ(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))+โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+๐›ผ(๐‘ก)๐‘‡๐‘˜(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+๐›ฝ(๐‘ก)๐‘‡๐‘™(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+โ„Ž1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+๐›ผ1(๐‘ก)๐‘‡๐‘˜1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)+๐›ฝ1(๐‘ก)๐‘‡๐‘™1๎€ป(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)๐‘‘๐‘ก,(10)subjecttoโˆ‡๐‘โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+โˆ‡๐‘๐›ผ(๐‘ก)๐‘‡๐‘˜(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+โˆ‡๐‘๐›ฝ(๐‘ก)๐‘‡โˆ‡๐‘™(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)=0,(11)๐‘žโ„Ž1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)+โˆ‡๐‘ž๐›ผ1(๐‘ก)๐‘‡๐‘˜1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)+โˆ‡๐‘ž๐›ฝ1(๐‘ก)๐‘‡๐‘™1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)=0,(12)๐‘ฆ(๐‘Ž)=๐›พ1,๐‘ฆ(๐‘)=๐›พ2;๐‘ฃ(๐‘Ž)=๐›ฟ1,๐‘ฃ(๐‘)=๐›ฟ2,(13)๐›ผ(๐‘ก)โˆˆโ„๐‘Ÿ+,๐›ฝ(๐‘ก)โˆˆโ„๐‘ ,๐›ผ1(๐‘ก)โˆˆโ„๐‘Ÿ1+,๐›ฝ1(๐‘ก)โˆˆโ„๐‘ 1,๐‘โˆˆโ„๐‘›,๐‘žโˆˆโ„๐‘š.(14)

Remark 3.1. If ๎‚ƒ๐‘“โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)=๐‘๐‘ฅ๐‘‘(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))โˆ’๐‘“๐‘‘๐‘กฬ‡๐‘ฅ๎‚„+1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))2๐‘๎‚ธ๐‘“๐‘ฅ๐‘ฅ๐‘‘(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))โˆ’2๐‘“๐‘‘๐‘ก๐‘ฅฬ‡๐‘ฅ(+๐‘‘๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))2๐‘‘๐‘ก2๐‘“ฬ‡๐‘ฅฬ‡๐‘ฅ๎‚นโ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))๐‘,1๎€บ๐‘“(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)=๐‘ž๐‘ข๎€ป+1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))2๐‘ž๎€บ๐‘“๐‘ข๐‘ข(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))+2๐‘“๐‘ฅ๐‘ข๎€ป๐‘˜(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))๐‘ž,๐‘—๎‚ƒ๐‘”(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)=๐‘๐‘—๐‘ฅ๐‘‘(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))โˆ’๐‘”๐‘‘๐‘ก๐‘—ฬ‡๐‘ฅ๎‚„+1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))2๐‘๎‚ธ๐‘”๐‘—๐‘ฅ๐‘ฅ๐‘‘(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))โˆ’2๐‘”๐‘‘๐‘ก๐‘—๐‘ฅฬ‡๐‘ฅ+๐‘‘(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))2๐‘‘๐‘ก2๐‘”๐‘—ฬ‡๐‘ฅฬ‡๐‘ฅ๎‚น๐‘˜(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))๐‘,๐‘—=1,2,3,โ€ฆ,๐‘Ÿ,๐‘—1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)=๐‘ž๐‘”๐‘—๐‘ข+1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))2๐‘ž๎€บ๐‘”๐‘—๐‘ข๐‘ข(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))+2๐‘”๐‘—๐‘ฅ๐‘ข๎€ป(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))๐‘ž,๐‘—=1,2,โ€ฆ,๐‘Ÿ1,๐‘™๐‘–๎‚ƒ๐บ(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)=๐‘๐‘–๐‘ฅ๐‘‘(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))โˆ’๐บ๐‘‘๐‘ก๐‘–ฬ‡๐‘ฅ๎‚„+1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))2๐‘๎‚ธ๐บ๐‘–๐‘ฅ๐‘ฅ๐‘‘(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))โˆ’2๐บ๐‘‘๐‘ก๐‘–๐‘ฅฬ‡๐‘ฅ(+๐‘‘๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))2๐‘‘๐‘ก2๐บ๐‘–ฬ‡๐‘ฅฬ‡๐‘ฅ๎‚น๐‘™(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))๐‘,๐‘–=1,2,3,โ€ฆ,๐‘ ,๐‘–1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)=๐‘ž๐บ๐‘–๐‘ข+1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))2๐‘ž๎€บ๐บ๐‘–๐‘ข๐‘ข(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))+2๐บ๐‘–๐‘ฅ๐‘ข(๎€ป๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))๐‘ž,๐‘–=1,2,3,โ€ฆ,๐‘ 1,(15) then (MHCD) is similar to the second-order duality given by Padhan and Nahak [10].

Theorem 3.2 (weak duality). Let (๐‘ฅ(๐‘ก),๐‘ข(๐‘ก)) and (๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐›ผ(๐‘ก),๐›ฝ(๐‘ก),๐‘,๐‘ž) be the feasible solutions of (CP) and (MHCD), respectively. Let โˆซ๐‘๐‘Ž๐‘“(๐‘ก,โ‹…,โ‹…,โ‹…)๐‘‘๐‘ก, โˆซ๐‘๐‘Ž๐›ผ(๐‘ก)๐‘‡๐‘”(๐‘ก,โ‹…,โ‹…,โ‹…)๐‘‘๐‘ก and โˆซ๐‘๐‘Ž๐›ฝ(๐‘ก)๐‘‡๐บ(๐‘ก,โ‹…,โ‹…,โ‹…)๐‘‘๐‘ก be higher-order ๐œŒ0โˆ’(๐œ‚,๐œ‰,๐œƒ)-invex, higher-order ๐œŒ1โˆ’(๐œ‚,๐œ‰,๐œƒ)-invex, and higher-order ๐œŒ2โˆ’(๐œ‚,๐œ‰,๐œƒ)-invex functions in ๐‘ฅ, ฬ‡๐‘ฅ, and ๐‘ข on ๐ผ with respect to the same functions ๐œ‚,๐œ‰,๐œƒ, with ๐œŒ0+๐œŒ1+๐œŒ2โ‰ฅ0, then the following inequality holds between the primal (CP) and the dual (MHCD): ๎€œ๐‘๐‘Žโ‰ฅ๎€œ๐‘“(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))๐‘‘๐‘ก๐‘๐‘Ž๎€บ๐‘“(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))+๐›ผ(๐‘ก)๐‘‡๐‘”(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))+๐›ฝ(๐‘ก)๐‘‡๐บ(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))+โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+โ„Ž1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)+๐›ผ(๐‘ก)๐‘‡๐‘˜(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+๐›ผ1(๐‘ก)๐‘‡๐‘˜1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)+๐›ฝ(๐‘ก)๐‘‡๐‘™(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+๐›ฝ1(๐‘ก)๐‘‡๐‘™1๎€ป(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)๐‘‘๐‘ก.(16)

Proof. Since (๐‘ฅ(๐‘ก),๐‘ข(๐‘ก)) and (๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐›ผ(๐‘ก),๐›ฝ(๐‘ก),๐‘,๐‘ž) be the feasible solutions of (CP) and (MHCD), respectively, we have ๎€œ๐‘๐‘Žโˆ’๎€œ๐‘“(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))๐‘‘๐‘ก๐‘๐‘Ž๎€บ๐‘“(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))+๐›ผ(๐‘ก)๐‘‡๐‘”(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))+๐›ฝ(๐‘ก)๐‘‡๐บ(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))+โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+โ„Ž1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)+๐›ผ(๐‘ก)๐‘‡๐‘˜(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+๐›ผ1(๐‘ก)๐‘‡๐‘˜1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)+๐›ฝ(๐‘ก)๐‘‡๐‘™(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+๐›ฝ1(๐‘ก)๐‘‡๐‘™1๎€ปโ‰ฅ๎‚ธ๎€œ(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)๐‘‘๐‘ก๐‘๐‘Žโˆ’๎€œ๐‘“(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))๐‘‘๐‘ก๐‘๐‘Ž๎‚น+๎‚ธ๎€œ๐‘“(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))๐‘‘๐‘ก๐‘๐‘Ž๐›ผ(๐‘ก)๐‘‡โˆ’๎€œ๐‘”(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))๐‘‘๐‘ก๐‘๐‘Ž๐›ผ(๐‘ก)๐‘‡๎‚น+๎‚ธ๎€œ๐‘”(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))๐‘‘๐‘ก๐‘๐‘Ž๐›ฝ(๐‘ก)๐‘‡๐บโˆ’๎€œ(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))๐‘‘๐‘ก๐‘๐‘Ž๐›ฝ(๐‘ก)๐‘‡๎‚นโˆ’๎€œ๐บ(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))๐‘‘๐‘ก๐‘๐‘Ž๎€บโ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+โ„Ž1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)+๐›ผ(๐‘ก)๐‘‡๐‘˜(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+๐›ผ1(๐‘ก)๐‘‡๐‘˜1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)+๐›ฝ(๐‘ก)๐‘‡๐‘™(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+๐›ฝ1(๐‘ก)๐‘‡๐‘™1๎€ป(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)๐‘‘๐‘กโ‰ฅ0.(17) (by the higher-order ๐œŒโˆ’(๐œ‚,๐œ‰,๐œƒ)-invexity of โˆซ๐‘๐‘Žโˆซ๐‘“(๐‘ก,โ‹…,โ‹…,โ‹…)๐‘‘๐‘ก,๐‘๐‘Ž๐›ผ(๐‘ก)๐‘‡๐‘”(๐‘ก,โ‹…,โ‹…,โ‹…)๐‘‘๐‘ก) and โˆซ๐‘๐‘Ž๐›ฝ(๐‘ก)๐‘‡๐บ(๐‘ก,โ‹…,โ‹…,โ‹…)๐‘‘๐‘ก,๐œŒ0+๐œŒ1+๐œŒ2โ‰ฅ0 and equations (11)โ€“(14)). This completes the proof.

We construct the following example which verifies Theorem 3.2 above, in which the objective and the constraints functions are higher-order ๐œŒโˆ’(๐œ‚,๐œ‰,๐œƒ)-invex.

Example 3.3. Let us consider the following control problem: ๎€œmin๐‘๐‘Ž๎€ท๐‘ฅ2๎€ธ(๐‘ก)โˆ’๐‘ฅ(๐‘ก)โˆ’๐‘ข(๐‘ก)๐‘‘๐‘ก,(18)subjectto๐‘ข(๐‘ก)โˆ’๐‘ฅ2(๐‘ก)โ‰ค0,๐‘ฅ(๐‘ก)โˆ’๐‘ข2(๐‘ก)=0,๐‘ฅ(๐‘Ž)=๐›พ1,๐‘ฅ(๐‘)=๐›พ2,๐‘ข(๐‘Ž)=๐›ฟ1,๐‘ข(๐‘)=๐›ฟ2,(19) where ๐ผ=[๐‘Ž,๐‘],๐‘“โˆถ๐ผร—โ„ร—โ„โ†’โ„,๐‘”โˆถ๐ผร—โ„ร—โ„โ†’โ„ and ๐บโˆถ๐ผร—โ„ร—โ„โ†’โ„, ๐‘“(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))=๐‘ฅ2(๐‘ก)โˆ’๐‘ฅ(๐‘ก)โˆ’๐‘ข(๐‘ก), ๐‘”(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))=๐‘ข(๐‘ก)โˆ’๐‘ฅ2(๐‘ก) and ๐บ(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))=๐‘ฅ(๐‘ก)โˆ’๐‘ข2(๐‘ก). It is clear that the objective function ๐‘“, the inequality constraint function ๐‘” and the equality constraint function ๐บ are not 1โˆ’(๐œ‚,๐œ‰,๐œƒ)-invex, โˆ’(1/2)โˆ’(๐œ‚,๐œ‰,๐œƒ)-invex and โˆ’(1/2)โˆ’(๐œ‚,๐œ‰,๐œƒ)-invex, respectively, as defined by Padhan and Nahak [10]. But for ๐‘ข(๐‘ก)๐‘ฃ(๐‘ก)>1 and ๐‘ฆ(๐‘ก)>๐‘ฃ(๐‘ก), they are higher-order 1โˆ’(๐œ‚,๐œ‰,๐œƒ)-invex, higher-order โˆ’(1/2)โˆ’(๐œ‚,๐œ‰,๐œƒ)-invex and higher-order โˆ’(1/2)โˆ’(๐œ‚,๐œ‰,๐œƒ)-invex, respectively, with respect to the same ๐œ‚,๐œ‰ and ๐œƒ. The functions ๐œ‚,๐œ‰,๐œƒโˆถ๐ผร—โ„ร—โ„ร—โ„ร—โ„ร—โ„ร—โ„โ†’โ„, โ„Žโˆถ๐ผร—โ„ร—โ„ร—โ„ร—โ„โ†’โ„, ๐‘˜โˆถ๐ผร—โ„ร—โ„ร—โ„ร—โ„โ†’โ„, ๐‘™โˆถ๐ผร—โ„ร—โ„ร—โ„ร—โ„โ†’โ„,โ„Ž1โˆถ๐ผร—โ„ร—โ„ร—โ„ร—โ„โ†’โ„, ๐‘˜1โˆถ๐ผร—โ„ร—โ„ร—โ„ร—โ„โ†’โ„ and ๐‘™1โˆถ๐ผร—โ„ร—โ„ร—โ„ร—โ„โ†’โ„ are defined as follows: ๎€ท๐‘ฅ๐œ‚(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ข(๐‘ก),๐‘ฃ(๐‘ก))=โˆ’2๎€ธ,๎€ท(๐‘ก)+๐‘ฅ(๐‘ก)+๐‘ข(๐‘ก)+๐‘ฆ(๐‘ก)๐œ‰(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ข(๐‘ก),๐‘ฃ(๐‘ก))=โˆ’2๐‘ฅ2๎€ธ,=โˆš(๐‘ก)๐‘ฆ(๐‘ก)+2๐‘ฅ(๐‘ก)๐‘ฆ(๐‘ก)+2๐‘ฆ(๐‘ก)๐‘ข(๐‘ก)โˆ’๐‘ข(๐‘ก)+๐‘ฃ(๐‘ก)๐œƒ(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ข(๐‘ก),๐‘ฃ(๐‘ก))๐‘ฅ2(๐‘ก)+๐‘ฆ2(๐‘ก),โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)=(2๐‘ฆ(๐‘ก)โˆ’1)๐‘โˆ’๐‘ฆ2(๐‘ก)โˆ’๐‘ฃ2(๐‘ก),๐‘˜(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)=โˆ’2๐‘ฆ(๐‘ก)๐‘โˆ’๐‘ฃ2(๐‘ก),๐‘™(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)=๐‘โˆ’๐‘ฆ2(๐‘ก)โˆ’๐‘ฃ2โ„Ž(๐‘ก),1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)=โˆ’๐‘žโˆ’๐‘ฆ2๐‘˜(๐‘ก),1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)=๐‘žโˆ’๐‘ฃ2๐‘™(๐‘ก),1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)=โˆ’๐‘ฆ2(๐‘ก)โˆ’๐‘ฃ2(๐‘ก).(20) The Mangasarian higher-order dual of the control problem is ๎€œmax๐‘๐‘Ž๎€ทโˆ’4๐‘ฆ2(๐‘ก)โˆ’6๐‘ฃ2๎€ธ(๐‘ก)๐‘‘๐‘ก,(21)subjectto๐‘ฆ(๐‘Ž)=๐›พ1,๐‘ฆ(๐‘)=๐›พ2;๐‘ฃ(๐‘Ž)=๐›ฟ1,๐‘ฃ(๐‘)=๐›ฟ2,๐›ผ(๐‘ก)=๐›ฝ(๐‘ก)=๐›ผ1(๐‘ก)=๐›ฝ1(๐‘ก)=1.(22) The above problem satisfies weak duality Theorem 3.2 for ๐‘ฅ2(๐‘ก)+4๐‘ฆ2(๐‘ก)+6๐‘ฃ2(๐‘ก)โ‰ฅ๐‘ฅ(๐‘ก)+๐‘ข(๐‘ก).

Necessary conditions for the existence of an extremal solution for a variational problem subject to the both equality and inequality constraints were given by Valentine [11]. Using Valentine's results, Berkovitz [12] obtained the corresponding necessary conditions for the control problem (CP). These may be stated in the following way. If (๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก)) is an optimal solution for (CP), then๐œ‡0๐‘“๐‘ฅ(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))+๐‘”๐‘ฅ(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))๐‘‡๐›ผ(๐‘ก)+๐บ๐‘ฅ(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))๐‘‡=๐‘‘๐›ฝ(๐‘ก)๎€ท๐œ‡๐‘‘๐‘ก0๐‘“ฬ‡๐‘ฅ(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))+๐‘”ฬ‡๐‘ฅ(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))๐‘‡๐›ผ(๐‘ก)+๐บฬ‡๐‘ฅ(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))๐‘‡๎€ธ,๐œ‡๐›ฝ(๐‘ก)(23)0๐‘“๐‘ข(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))+๐‘”๐‘ข(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))๐‘‡๐›ผ(๐‘ก)+โ„Ž๐‘ข(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))๐‘‡๐›ฝ(๐‘ก)=0,(24)๐›ผ(๐‘ก)๐‘‡๐‘”(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))=0,(25)๐›ฝ(๐‘ก)๐‘‡๐บ(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))=0,(26)๐›ผ(๐‘ก)โˆˆโ„๐‘Ÿ+(27) hold throughout ๐‘Žโ‰ค๐‘กโ‰ค๐‘ (except for the values of ๐‘ก corresponding to points of discontinuity of ๐‘ข(๐‘ก), (23) holds for right and left hand limits). Here, ๐œ‡0 is nonnegative constant, ๐›ฝ(๐‘ก) is continuous in ๐‘Žโ‰ค๐‘กโ‰ค๐‘, and ๐œ‡0,๐›ผ(๐‘ก), and ๐›ฝ(๐‘ก) cannot vanish simultaneously for any ๐‘Žโ‰ค๐‘กโ‰ค๐‘. It will be assumed that the minimizing arc determined by ๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก) is normal, that is, that ๐œ‡0 can be taken equal to 1.

Theorem 3.4 (strong duality). Let (๐‘ฅ(๐‘ก),๐‘ข(๐‘ก)) be a local or global optimal solution of (๐ถ๐‘ƒ) at which the constraint qualification (23)โ€“(27) are satisfied, and for the piecewise smooth functions ๐›ผโˆถ๐ผโ†’โ„๐‘Ÿ, ๐›ฝโˆถ๐ผโ†’โ„๐‘ , let (i)โ„Ž(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),0)=โ„Ž1(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),0)=0,(ii)๐‘˜(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),0)=๐‘˜1(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),0)=0,(iii)๐‘™(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),0)=๐‘™1(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),0)=0,(iv)โˆ‡๐‘โ„Ž(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),0)=๐‘“๐‘ฅ(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))โˆ’(๐‘‘/๐‘‘๐‘ก)๐‘“ฬ‡๐‘ฅ(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก)),(v)โˆ‡๐‘๐›ผ(๐‘ก)๐‘‡๐‘˜(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),0) = ๐‘”๐‘ฅ(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))๐‘‡๐›ผ(๐‘ก)โˆ’(๐‘‘/๐‘‘๐‘ก)๐‘”ฬ‡๐‘ฅ(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))๐‘‡๐›ผ(๐‘ก),(vi)โˆ‡๐‘๐›ฝ(๐‘ก)๐‘‡๐‘™(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),0)= ๐บ๐‘ฅ(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))๐‘‡๐›ฝ(๐‘ก)โˆ’(๐‘‘/๐‘‘๐‘ก)๐บฬ‡๐‘ฅ(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))๐‘‡๐›ฝ(๐‘ก), (vii)โˆ‡๐‘žโ„Ž1(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),0)=๐‘“๐‘ข(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก)),(viii)โˆ‡๐‘ž๐›ผ(๐‘ก)๐‘‡๐‘˜1(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),0)=๐‘”๐‘ข(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))๐‘‡๐›ผ(๐‘ก),(ix)โˆ‡๐‘ž๐›ฝ(๐‘ก)๐‘‡๐‘™1(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),0)= ๐บ๐‘ข(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))๐‘‡๐›ฝ(๐‘ก).Then, (๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),๐›ผ(๐‘ก),๐›ฝ(๐‘ก),๐‘=0,๐‘ž=0) is feasible for (MHCD). Moreover, if the weak duality Theorem 3.2 holds between the control primal (๐ถ๐‘ƒ) and the Mangasarian higher-order dual (MHCD). Then, (๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),๐›ผ(๐‘ก),๐›ฝ(๐‘ก),๐‘=0,๐‘ž=0) is an optimal solution of (MHCD), and the optimal values of (CP) and (MHCD) are equal.

Proof. Since (๐‘ฅ(๐‘ก),๐‘ข(๐‘ก)) is an optimal solution of (CP), from (23)โ€“(27), we can easily conclude that (๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),๐›ผ(๐‘ก),๐›ฝ(๐‘ก),๐‘=0,๐‘ž=0) satisfies the constraints of (MHCD) and objective values of (CP) and (MHCD) are equal. Hence, the result follows.

Theorem 3.5 (converse duality). Let (๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),๐›ผ(๐‘ก),๐›ฝ(๐‘ก),๐‘,๐‘ž) be an optimal solution of (MWHCD). Suppose that โˆซ๐‘๐‘Ž๐‘“(๐‘ก,โ‹…,โ‹…,โ‹…)๐‘‘๐‘ก, โˆซ๐‘๐‘Ž๐›ผ(๐‘ก)๐‘‡๐‘”(๐‘ก,โ‹…,โ‹…,โ‹…)๐‘‘๐‘ก and โˆซ๐‘๐‘Ž๐›ฝ(๐‘ก)๐‘‡๐บ(๐‘ก,โ‹…,โ‹…,โ‹…)๐‘‘๐‘ก are higher-order ๐œŒ0โˆ’(๐œ‚,๐œ‰,๐œƒ)-invex, higher-order ๐œŒ1โˆ’(๐œ‚,๐œ‰,๐œƒ)-invex, and higher-order ๐œŒ2โˆ’(๐œ‚,๐œ‰,๐œƒ)-invex functions in ๐‘ฅ, ฬ‡โ€Œ๐‘ฅ, and ๐‘ข on ๐ผ with respect to the same functions ๐œ‚,๐œ‰,๐œƒ, with ๐œŒ0+๐œŒ1+๐œŒ2โ‰ฅ0. Moreover, if ๐›ผ(๐‘ก)๐‘‡๐‘”๎€ท๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๎€ธ+๐‘ข(๐‘ก)๐›ฝ(๐‘ก)๐‘‡๐บ๎€ท๐‘ก,๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๎€ธ๎‚€๐‘ข(๐‘ก)ร—โ„Ž๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),๐‘๎‚+โ„Ž1๎‚€๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),๐‘ž๎‚+๎‚€๐›ผ๐‘˜๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),๐‘๎‚+๐›ผ1๐‘˜1๎‚€๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),๐‘ž๎‚+๎‚€๐›ฝ๐‘™๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),๐‘๎‚+๐›ฝ1๐‘™1๎‚€๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),๐‘ž๎‚โ‰ฅ0,(28) then (๐‘ฅ(๐‘ก),๐‘ข(๐‘ก)) is an optimal solution of (CP).

Proof. Suppose that (๐‘ฅ(๐‘ก),๐‘ข(๐‘ก)) is not an optimal solution of (CP). Then, there exists a feasible solution (๐‘ฅ(๐‘ก),๐‘ข(๐‘ก)) of the primal (CP) such that ๎€œ๐‘๐‘Ž๎€œ๐‘“(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))๐‘‘๐‘ก<๐‘๐‘Ž๐‘“๎‚€๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๎‚๐‘ข(๐‘ก)๐‘‘๐‘ก.(29) Sinceโˆซ๐‘๐‘Ž๐‘“(๐‘ก,โ‹…,โ‹…,โ‹…)๐‘‘๐‘ก, โˆซ๐‘๐‘Ž๐›ผ(๐‘ก)๐‘‡๐‘”(๐‘ก,โ‹…,โ‹…,โ‹…)๐‘‘๐‘ก and โˆซ๐‘๐‘Ž๐›ฝ(๐‘ก)๐‘‡๐บ(๐‘ก,โ‹…,โ‹…,โ‹…)๐‘‘๐‘ก are higher-order ๐œŒ0โˆ’(๐œ‚,๐œ‰,๐œƒ)-invex, higher-order ๐œŒ1โˆ’(๐œ‚,๐œ‰,๐œƒ)-invex, and higher-order ๐œŒ2โˆ’(๐œ‚,๐œ‰,๐œƒ)-invex functions with respect to same ๐œ‚, ๐œ‰ and ๐œƒ, we have ๎€œ๐‘๐‘Ž๎€œ๐‘“(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))๐‘‘๐‘กโˆ’๐‘๐‘Ž๐‘“๎‚€๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๎‚โ‰ฅ๎€œ๐‘ข(๐‘ก)๐‘‘๐‘ก๐‘๐‘Ž๎‚ƒ๐›ผ(๐‘ก)๐‘‡๐‘”๎‚€๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๎‚+๐‘ข(๐‘ก)๐›ฝ(๐‘ก)๐‘‡๐บ๎‚€๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๎‚๎‚€๐‘ข(๐‘ก)+โ„Ž๐‘ก,๐‘ฅฬ‡(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),๐‘๎‚+โ„Ž1๎‚€๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),๐‘ž๎‚+๎‚€๐›ผ๐‘˜๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),๐‘๎‚+๐›ผ1๐‘˜1๎‚€๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),๐‘ž๎‚+๎‚€๐›ฝ๐‘™๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),๐‘๎‚+๐›ฝ1๐‘™1๎‚€๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),๐‘ž๎‚๎‚„๐‘‘๐‘ก(byequations(2),(3),(11),(12),and(14))โ‰ฅ0(byequations(28)).(30) This completes the proof.

Mond-Weir type higher-order duality is established to weaken the higher-order invexity requirements, that is, higher-order pseudoinvexiy and higher-order quasi-invexity.

4. Mond-Weir Type Higher-Order Duality

In this section, we propose the following Mond-Weir type higher-order dual (MWHCD) to (CP): ๎€œ(MWHCD)max๐‘๐‘Ž๎€บ๐‘“(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))+โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+โ„Ž1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)โˆ’๐‘๐‘‡โˆ‡๐‘โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)โˆ’๐‘ž๐‘‡โˆ‡๐‘žโ„Ž1๎€ป(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)๐‘‘๐‘ก,(31)subjecttoโˆ‡๐‘โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+โˆ‡๐‘๐›ผ(๐‘ก)๐‘‡๐‘˜(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+โˆ‡๐‘๐›ฝ(๐‘ก)๐‘‡โˆ‡๐‘™(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)=0,(32)๐‘žโ„Ž1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)+โˆ‡๐‘ž๐›ผ1(๐‘ก)๐‘‡๐‘˜1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)+โˆ‡๐‘ž๐›ฝ1(๐‘ก)๐‘‡๐‘™1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)=0,(33)๐›ผ(๐‘ก)๐‘‡๐‘”(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))+๐›ผ(๐‘ก)๐‘‡๐‘˜(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+๐›ผ1(๐‘ก)๐‘‡๐‘˜1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)โˆ’๐‘๐‘‡โˆ‡๐‘๐›ผ(๐‘ก)๐‘‡๐‘˜(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)โˆ’๐‘ž๐‘‡โˆ‡๐‘ž๐›ผ1(๐‘ก)๐‘‡๐‘˜1๐›ฝ(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)โ‰ฅ0,(34)1(๐‘ก)๐‘‡๐บ(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))+๐›ฝ(๐‘ก)๐‘‡๐‘™(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+๐›ฝ1(๐‘ก)๐‘‡๐‘™1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)โˆ’๐‘๐‘‡โˆ‡๐‘๐›ฝ(๐‘ก)๐‘‡๐‘™(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)โˆ’๐‘ž๐‘‡โˆ‡๐‘ž๐›ฝ1(๐‘ก)๐‘‡๐‘™1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)โ‰ฅ0,(35)๐‘ฆ(๐‘Ž)=๐›พ1,๐‘ฆ(๐‘)=๐›พ2,๐‘ฃ(๐‘Ž)=๐›ฟ1,๐‘ฃ(๐‘)=๐›ฟ2,(36)๐›ผ(๐‘ก)โˆˆโ„๐‘Ÿ+,๐›ฝ(๐‘ก)โˆˆโ„๐‘ ,๐›ผ1(๐‘ก)โˆˆโ„๐‘Ÿ1+,๐›ฝ1(๐‘ก)โˆˆโ„๐‘ 1,๐‘โˆˆโ„๐‘›,๐‘žโˆˆโ„๐‘š.(37)

Theorem 4.1 (weak duality). Let (๐‘ฅ(๐‘ก),๐‘ข(๐‘ก)) and (๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐›ผ(๐‘ก),๐›ฝ(๐‘ก),๐‘,๐‘ž) be the feasible solutions of (CP) and (MWHCD), respectively. Let โˆซ๐‘๐‘Ž๐‘“(๐‘ก,โ‹…,โ‹…,โ‹…)๐‘‘๐‘ก, โˆซ๐‘๐‘Ž๐›ผ(๐‘ก)๐‘‡๐‘”(๐‘ก,โ‹…,โ‹…,โ‹…)๐‘‘๐‘ก and โˆซ๐‘๐‘Ž๐›ฝ(๐‘ก)๐‘‡๐บ(๐‘ก,โ‹…,โ‹…,โ‹…)๐‘‘๐‘ก be higher-order ๐œŒ0โˆ’(๐œ‚,๐œ‰,๐œƒ)-pseudoinvex, higher-order ๐œŒ1โˆ’(๐œ‚,๐œ‰,๐œƒ)-quasi-invex, and higher-order ๐œŒ2โˆ’(๐œ‚,๐œ‰,๐œƒ)-quasi-invex functions in ๐‘ฅ, ฬ‡๐‘ฅ, and ๐‘ข on ๐ผ with respect to the same functions ๐œ‚,๐œ‰, and ๐œƒ, with ๐œŒ0+๐œŒ1+๐œŒ2โ‰ฅ0, then the following inequality holds between the primal (CP) and the dual (MHCD), ๎€œ๐‘๐‘Žโ‰ฅ๎€œ๐‘“(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))๐‘‘๐‘ก๐‘๐‘Ž๎€บ๐‘“(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))+โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+โ„Ž1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)โˆ’๐‘๐‘‡โˆ‡๐‘โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)โˆ’๐‘ž๐‘‡โˆ‡๐‘žโ„Ž1๎€ป(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)๐‘‘๐‘ก.(38)

Proof. Since (๐‘ฅ(๐‘ก),๐‘ข(๐‘ก)) and (๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐›ผ(๐‘ก),๐›ฝ(๐‘ก),๐‘,๐‘ž) are the feasible solutions of (CP) and (MWHCD), respectively, from (46), (3), (34), (35) and (37), we have ๐›ผ(๐‘ก)๐‘‡๐‘”(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))โˆ’๐›ผ(๐‘ก)๐‘‡๐‘”(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))โˆ’๐›ผ(๐‘ก)๐‘‡๐‘˜(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)โˆ’๐›ผ1(๐‘ก)๐‘‡๐‘˜1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)+๐‘๐‘‡โˆ‡๐‘๐›ผ(๐‘ก)๐‘‡๐‘˜(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+๐‘ž๐‘‡โˆ‡๐‘ž๐›ผ1(๐‘ก)๐‘‡๐‘˜1๐›ฝ(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)โ‰ค0,(39)1(๐‘ก)๐‘‡๐บ(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))โˆ’๐›ฝ1(๐‘ก)๐‘‡๐บ(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))โˆ’๐›ฝ(๐‘ก)๐‘‡๐‘™(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)โˆ’๐›ฝ1(๐‘ก)๐‘‡๐‘™1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)+๐‘๐‘‡โˆ‡๐‘๐›ฝ(๐‘ก)๐‘‡๐‘™(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+๐‘ž๐‘‡โˆ‡๐‘ž๐›ฝ1(๐‘ก)๐‘‡๐‘™1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)โ‰ค0.(40) Since โˆซ๐‘๐‘Ž๐›ผ(๐‘ก)๐‘‡๐‘”(๐‘ก,โ‹…,โ‹…,โ‹…)๐‘‘๐‘ก and โˆซ๐‘๐‘Ž๐›ฝ(๐‘ก)๐‘‡๐บ(๐‘ก,โ‹…,โ‹…,โ‹…)๐‘‘๐‘ก are higher-order ๐œŒ1โˆ’(๐œ‚,๐œ‰,๐œƒ)-quasi-invex and higher-order ๐œŒ2โˆ’(๐œ‚,๐œ‰,๐œƒ)-quasi-invex functions, (39) and (40) gives ๎€œ๐‘๐‘Ž๎€บ๐œ‚(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ข(๐‘ก),๐‘ฃ(๐‘ก))๐‘‡๎€ทโˆ‡๐‘๐›ผ(๐‘ก)๐‘‡๐‘˜(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+โˆ‡๐‘๐›ฝ(๐‘ก)๐‘‡๎€ธ๐‘™(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+๐œ‰(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ข(๐‘ก),๐‘ฃ(๐‘ก))๐‘‡+๎€ทโˆ‡๐‘ž๐›ผ1(๐‘ก)๐‘‡๐‘˜1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)+โˆ‡๐‘ž๐›ฝ1(๐‘ก)๐‘‡๐‘™1๎€ธ+๎€ท๐œŒ(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)1+๐œŒ2๎€ธ(โ€–๐œƒ๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ข(๐‘ก),๐‘ฃ(๐‘ก))โ€–2๎€ปโŸน๎€œ๐‘‘๐‘กโ‰ค0๐‘๐‘Ž๎€บ๐œ‚(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ข(๐‘ก),๐‘ฃ(๐‘ก))๐‘‡โˆ‡๐‘โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+๐œ‰(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ข(๐‘ก),๐‘ฃ(๐‘ก))๐‘‡โˆ‡๐‘žโ„Ž1โ€–(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)+๐œŒโ€–๐œƒ(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ข(๐‘ก),๐‘ฃ(๐‘ก))2๎€ป๎€ท๐‘‘๐‘กโ‰ฅ0byequations(32),(33)and๐œŒ0+๐œŒ1+๐œŒ2๎€ธโŸน๎€œโ‰ฅ0๐‘๐‘Ž๐‘“โˆ’๎€œ(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))๐‘‘๐‘ก๐‘๐‘Ž๎€บ๐‘“(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก))+โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)+โ„Ž1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)โˆ’๐‘๐‘‡โˆ‡๐‘โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)โˆ’๐‘ž๐‘‡โˆ‡๐‘žโ„Ž1๎€ป(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)๐‘‘๐‘กโ‰ฅ0.(41) by the higher-order โ€‰๐œŒ0โˆ’(๐œ‚,๐œ‰,๐œƒ)-pseudoinvexity ofโ€‰โˆซ๐‘๐‘Ž๐‘“(๐‘ก,โ‹…,โ‹…,โ‹…)๐‘‘๐‘ก. This completes the proof.

We construct the following example which verifies Theorem 4.1 above, in which the objective function is higher-order ๐œŒโˆ’(๐œ‚,๐œ‰,๐œƒ)-pseudoinvex and the constraints functions are higher-order ๐œŒโˆ’(๐œ‚,๐œ‰,๐œƒ)-quasi-invex.

Example 4.2. Let us consider the following control problem: ๎€œmin๐‘๐‘Ž๎€ท๐‘ฅ2(๐‘ก)+๐‘ข2๎€ธ(๐‘ก)+๐‘ฅ(๐‘ก)+๐‘ข(๐‘ก)๐‘‘๐‘ก,(42)subjectto๐‘ฅ2๐‘ข(๐‘ก)โˆ’๐‘ฅ(๐‘ก)๐‘ข(๐‘ก)โ‰ค0,2(๐‘ก)โˆ’๐‘ฅ(๐‘ก)=0,๐‘ฅ(๐‘Ž)=๐›พ1,๐‘ฅ(๐‘)=๐›พ2,๐‘ข(๐‘Ž)=๐›ฟ1,๐‘ข(๐‘)=๐›ฟ2,(43) where ๐ผ=[๐‘Ž,๐‘],๐‘“โˆถ๐ผร—โ„ร—โ„โ†’โ„,๐‘”โˆถ๐ผร—โ„ร—โ„โ†’โ„,๐บโˆถ๐ผร—โ„ร—โ„โ†’โ„, ๐‘“(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))=๐‘ฅ2(๐‘ก)+๐‘ข2(๐‘ก)+๐‘ฅ(๐‘ก)+๐‘ข(๐‘ก), ๐‘”(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))=๐‘ฅ2(๐‘ก)โˆ’๐‘ฅ(๐‘ก)๐‘ข(๐‘ก), and ๐บ(๐‘ก,๐‘ฅ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))=๐‘ข2(๐‘ก)โˆ’๐‘ฅ(๐‘ก). It is clear that the objective function ๐‘“, the inequality constraint function ๐‘” and the equality constraint function ๐บ are not higher-order -(1/2)-(๐œ‚,๐œ‰,๐œƒ)-invex, higher-order 1โˆ’(๐œ‚,๐œ‰,๐œƒ)โˆ’invex and higher-order -(1/2)-(๐œ‚,๐œ‰,๐œƒ)-invex, respectively. But they are higher-order -(1/2)-(๐œ‚,๐œ‰,๐œƒ)-pseudoinvex, higher-order 1-(๐œ‚,๐œ‰,๐œƒ)-quasi-invex and higher-order -(1/2)-(๐œ‚,๐œ‰,๐œƒ)-quasi-invex, respectively, with respect to the same ๐œ‚,๐œ‰ and ๐œƒ. The functions ๐œ‚,๐œ‰,๐œƒโˆถ๐ผร—โ„ร—โ„ร—โ„ร—โ„ร—โ„ร—โ„โ†’โ„,โ„Žโˆถ๐ผร—โ„ร—โ„ร—โ„ร—โ„โ†’โ„, ๐‘˜โˆถ๐ผร—โ„ร—โ„ร—โ„ร—โ„โ†’โ„, ๐‘™โˆถ๐ผร—โ„ร—โ„ร—โ„ร—โ„โ†’โ„,โ„Ž1โˆถ๐ผร—โ„ร—โ„ร—โ„ร—โ„โ†’โ„,๐‘˜1โˆถ๐ผร—โ„ร—โ„ร—โ„ร—โ„โ†’โ„, and ๐‘™1โˆถ๐ผร—โ„ร—โ„ร—โ„ร—โ„โ†’โ„ are defined as follows: ๎€ท๐‘ฅ๐œ‚(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ข(๐‘ก),๐‘ฃ(๐‘ก))=โˆ’2๎€ธ,๎€ท๐‘ข(๐‘ก)+๐‘ข(๐‘ก)+1๐œ‰(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ข(๐‘ก),๐‘ฃ(๐‘ก))=โˆ’2๎€ธ,=โˆš(๐‘ก)+๐‘ฅ(๐‘ก)+1๐œƒ(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก),ฬ‡๐‘ฅ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ข(๐‘ก),๐‘ฃ(๐‘ก))๐‘ฅ2(๐‘ก)๐‘ฆ2(๐‘ก)+๐‘ข(๐‘ก)๐‘ฆ2(๐‘ก)+๐‘ข2(๐‘ก)๐‘ฃ2(๐‘ก)+๐‘ฅ(๐‘ก)๐‘ฃ2๎€ท๐‘ฆ(๐‘ก),โ„Ž(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)=โˆ’2(๐‘ก)+๐‘ฃ2๎€ท๐‘ฆ(๐‘ก)โˆ’2(๐‘ก)+๐‘ฃ2๎€ธ๐‘๎€ธ,(๐‘ก)+1๐‘˜(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘)=๐‘ฆ(๐‘ก)๐‘ฃ(๐‘ก)+๐‘ฃ2(๐‘ก)+๐‘ฆ2โ„Ž(๐‘ก)๐‘,1๎€ท(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)=โˆ’2๐‘ฃ2๎€ท๐‘ฆ(๐‘ก)+๐‘ฆ(๐‘ก)+2(๐‘ก)+๐‘ฃ2๎€ธ๐‘ž๎€ธ,โ„Ž(๐‘ก)+11๎€ท(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)=โˆ’2๐‘ฃ2(๎€ท๐‘ฆ๐‘ก)+๐‘ฆ(๐‘ก)+2(๐‘ก)+๐‘ฃ2(๎€ธ๐‘ž๎€ธ,๐‘˜๐‘ก)+11(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)=๐‘ฆ2(๐‘ก)+๐‘ฃ2๐‘™(๐‘ก)๐‘ž,1(๐‘ก,๐‘ฆ(๐‘ก),ฬ‡๐‘ฆ(๐‘ก),๐‘ฃ(๐‘ก),๐‘ž)=4๐‘ฆ2(๐‘ก)+5๐‘ฃ2๎€ท๐‘ฆ(๐‘ก)+2๎€ธ(๐‘ก)+1๐‘ž.(44) The Mond-Weir higher-order dual of the control problem is ๎€œmax๐‘๐‘Ž๎€ท๐‘ฆ2(๐‘ก)+๐‘ฃ2๎€ธ(๐‘ก)๐‘‘๐‘ก,(45)subjectto2๐‘ฆ2(๐‘ก)+๐‘ฃ2(๐‘ก)โ‰ฅ0,(46)6๐‘ฆ2(๐‘ก)+9๐‘ฃ2(๐‘ก)โ‰ฅ0,(47)๐‘ฆ(๐‘Ž)=๐›พ1,๐‘ฆ(๐‘)=๐›พ2,๐‘ฃ(๐‘Ž)=๐›ฟ1,๐‘ฃ(๐‘)=๐›ฟ2,(48)๐›ผ(๐‘ก)=๐›ฝ(๐‘ก)=๐›ผ1(๐‘ก)=๐›ฝ1(๐‘ก)=1.(49) The above problem satisfies weak duality Theorem 4.1.

Theorem 4.3 (strong duality). Let (๐‘ฅ(๐‘ก),๐‘ข(๐‘ก)) be a local or global optimal solution of (CP), and for the piecewise smooth functions ๐›ผโˆถ๐ผโ†’โ„๐‘Ÿ, ๐›ฝโˆถ๐ผโ†’โ„๐‘ , let (i)โ„Ž(๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),0)=โ„Ž1(๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),0)=0,(ii)๐‘˜(๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),0)=๐‘˜1(๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),0)=0,(iii)๐‘™(๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),0)=๐‘™1(๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),0)=0,(iv)โˆ‡๐‘โ„Ž(๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),0)=๐‘“๐‘ฅ(๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก))โˆ’(๐‘‘/๐‘‘๐‘ก)๐‘“ฬ‡๐‘ฅ(๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก)),(v)โˆ‡๐‘๐›ผ(๐‘ก)๐‘‡๐‘˜(๐‘ก,ฬ‡๐‘ฅ(๐‘ก),๐‘ฅ(๐‘ก),๐‘ข(๐‘ก),0) = ๐‘”๐‘ฅ(๐‘ก