International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3441-3448 © Research India Publications. http://www.ripublication.com Unravelling Non-Differentiable Manifold Problems based on Lagrange and Wolfe Duality

Ganesh Kumar Thakur1, Bandana Priya2 and Sudesh Garg3 1Department of Mathematics, Krishna Engineering College, Ghaziabad, India. 2,3Department of Mathematics, G.L. Bajaj Institute of Technology & Management, Greater Noida, India.

Abstract pseudo boncavity assumptions [9]. A pair of multi objective second order symmetric dual nonlinear programming This paper presented a solution for solving the problems in problems, second order pseudo-invexity assumptions on the duality theorems for the category of non-differentiable multi- functions are involved in arbitrary cones. Subsequently, objective programming issues. Weak and weak, strong, converse and self-duality theorems are theorems solve optimization problem based on the established under second order pseudo-invexity or second formulation of the primal and dual problems. Here the order pseudo-incavity assumptions [10]. A weak second-order solution concepts of the primal and dual problems are based sufficient condition holds, and the feasible set is polyhedric, on the concept of Hybridization of both Lagrange and Wolfe formula for computing the directional derivative of the duality theorem. The proof are designed in such a way that optimal control with respect to a perturbation. [11] This is a the solution for the primal problem must always be greater partial extension of the results of [11] to the two norms than or equal to the solution of the dual problem. setting, and is to be compared with [12] and [13], where the Consequently the concepts of without in the weak derivative of solution is computed under stronger second- and strong sense are also introduced, and strong duality order conditions, whereas our theory of second-order theorems in the weak and strong sense are then derived. necessary conditions guaranties the fact that our sufficient Keywords: Second order duality, multi-objective condition is minimal [14]. programming issues, Type I function, Lagrange principle,

Wolfe duality. RELATED RESEARCHES AMS 2002 Subject Classification: 90C29, 90C30, 90C46. Xin-Min Yang D, et al. [15] have described a technique of

second-order symmetric dual models for multi objective INTRODUCTION nonlinear programming. Where, weak, strong, and converse duality theorems formulated, second-order symmetric dual Srivastava and Govil formulated second order Mond-Weir programs under invexity conditions was used. A pair of type dual for multi objective nonlinear programming and second-order nonlinear multi objective programs was established duality results used second order (F,ρ,σ) type-I formulated the primal problem (MSP) and the dual problem functions and their generalizations. Higher order cone convex, (MSD) were formed. Weak duality was feasible for MSP and pseudo convex and similar convex functions are studied and strong duality feasible for MSD. There the objective values of better order duality results for a vertex MSP and MSD were equal, efficient solution of MSD were improvement drawback over cones victimization higher order proved. The MSD efficient solution was used in converse has been introduced [1]. The convexity theory plays an duality. important role in many aspects of mathematical programming. In recent years, in order to relax convexity assumption, TANG LiPing, et al. [16] have described the weak duality various generalized convexity notions have been obtained. solution, where objective value of a feasible solution to the One of them is the concept of (F,  ) convex functions defined primal problem was not less than the corresponding dual one. by Preda[2], which extended the class of F- convex functions This solution provides a lower bound for the primal optimal and the classes of  convex functions defined by Vial [3,4]. value that was feasible dual solution. The strong duality that extended the category of F convex functions and also the theorem described, whenever the primal problem had an categories of  convex functions [5]. optimal solution, the dual problem also had one and there was no duality gap. However, the most difficult part of the duality The study of second order duality is important because theory, was on the converse duality theorem. It deals with the of the process advantage over initial order duality because issues on how to obtain the primal solution from the dual it provides tighter bounds for the worth of the objective solution and on conditions under which there is no gap function. Mangasarian [6] thought of a nonlinear program between the primal problem and the dual problem. and mentioned second order duality using certain inequalities. The conception of second order that was S.J. Lia, et al. [17] have described, Weir and Mond named as bonvex function by Bector and Chandra [7]. Later method that explain weak, strong and converse duality for Bector and Chandra [8] established second order symmetric weak minima of multiple objective optimization problems duality results for a combination under different pseudo-convexity and quasi convexity of nonlinear programming issues by pseudo bonvexity and assumptions. Mond–Weir type of duality results was

3441 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3441-3448 © Research India Publications. http://www.ripublication.com differentials of generalized d-type-I functions that involved in variables under the derived constraints on the dual variables. the multi objective programming problem. In that Mond–Weir duality scheme for optimization problems involving set functions, that defined on a measure space with the variables Primal and dual problem being measurable sets .In Mond–Weir type of duality results under generalized pseudo convexity and generalized quasi Let us consider the non- problem, convexity assumptions were investigated on duality properties Minimize f a *  (1) of optimization problems with set-valued mappings that satisfy an invex property and by virtue of tangent derivative of Subject to: g a*  0 for x 1,2,...... i set-valued mapping. x   Anurag Jayswal, et al. [18] Duality is one of the most * hx a   0 for x 1,2,...... j prominent in operation research that helps generating useful insights about the optimization problem. Second order duality a*  A has even greater significance over first order duality, since it provides tighter bounds for the value of the objective function The Lagrangian dual problem for non-linear programming when approximations were used, because it involves more problem is defined as, parameters. Another advantage of second order duality have been defined feasible point in the primal, where, first order Maximize L m,n (2) duality conditions do not apply, then second order duality used to provide a lower bound to the value of the primal Subject to: m  0 problem. Where,

Mishra SK, [19] have described second order symmetric i j * * * * (3) duality, under second order F-convexity F-concavity/second L m,n  inf{ f a  mx g x a  nx hx a : a  A order F-pseudo convexity F-pseudo concavity for second x1 x1 order Wolfe and Mond-Weir type models, respectively. These second order duality results are then used to formulate Wolfe type, and Mond-Weir type second order minimax mixed Theorem 1: Weak duality integer dual programs and a symmetric duality theorem is Consider the primal problem P given by eqn. (1) and its established under separability and second order F-convexity * F-concavity of the kernel function. Fractional symmetric Lagrangian dual problem D given by eqn. (2). Let a be a duality results and self-duality have also been discussed in * * * feasible solution to P that is, a  A g a  0 and h a  0. second order duality.     Also let m,nbe a feasible solution to D; that is m  0 .

Then, HYBRID LAGRANGE-WOLFE DUALITY FOR * MULTI-OBJECTIVE OPTIMIZATION PROBLEM f a  L m,n Duality is an important feature of optimization problem. There Proof: are several types of problems such as convex, linear, constrained, equality etc. By the employment of lagrangian We use the definition ofL given in eqn.(3) and the facts theorem, convex, inequality, concave problems can be solved * * * and by utilizing Wolfe theorem invexity problems can be that a  A , m  0 ga  0 and ha  0 solved. Therefore by the hybridization of lagrange and wolfe To prove: duality theorems all these problems can be solved. As this hybrid duality theorem solves all these problems that directs i j * * * * to the solution for multi-objective optimization problem. L m,n  f a  mx ga  nx ha  f a  x1 x1

Corollary: LAGRANGE DUALITY THEOREM * * * * Generally the term "dual problem" refers to Lagrangian dual inf{ f (a ) : a  A, g(a )  0,h(a )  0}  Sup{L (m,n) : m  0} problem but other dual problems like Wolfe dual problem and Fenchel dual problem can also be used. The Lagrangian 1: problem is acquired by creating the Lagrangian, using nonnegative Lagrange multipliers to add the constraints to the g(a* )  0,h(a* )  0 , for some a*  A,has no solution. objective function, and then solving for some primal variable values that minimize the Lagrangian. This solution obtains Define the set primal variables as functions of Lagrange multipliers, which * * * is called as dual variables, so that the new problem is to S  {(p,q,r) : p  (a ),q  g(a ),r  h(a ) , for some maximize the objective function with respect to the dual a*  A

3442 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3441-3448 © Research India Publications. http://www.ripublication.com

The set S is convex, since A, and g are convex and h is Theorem 2:Strong duality affine. Since System 1 has no solution, we have n n Let A be a nonempty in R .Let f : R  R that (0,0,0)  S . n m and g : R  R be convex and h : Rn  Rl be affine. Suppose that the following constraint qualification is satisfied. Recall the following corollary of the Supporting Hyper There exists an aˆ*  A such that g(aˆ* )  0 and plane Theorem: * h(aˆ )  0 and 0int h(a* ) where Corollary h(A)  {h(a* ) : a*  A}.Then, Let S be a nonempty convex set in R n and a * int S . Then U * * inf{(f a**** ): a A ,( g a ) 0,( h a ) there is a nonzero vector p such that p (a  a )  0 for (5) * each a cl . 0}  sup{L (m , n ) : m  0}, We then have, from the above corollary, that there exists a Where * U * U * * nonzero vector (m0 ,m ,n ) such that L (m,n)  inf{ f (a )  m g(a )  n h(a ) : a  A}. Furthermore, if the inf is finite, then U (m0 , m , n )[( p , q ,) r  (0,0,0)] (4) sup{ (m,n) : m  0 , is achieved at m,n with m  0 if UU L m p  m q  n r  0 U * 0 the is achieved at a * then m g(a )  0 For each ( p,q,r) cl . Let  inf{ f (a* ) : a*  A, g(a* )  0,h(a* )  0 . * Now, fix an a  A . Noticing, from the definition of , that * p and q can be made arbitrarily large, we have that in order By assumption there exists a feasible solution aˆ for the primal problem and hence  . to satisfy (4), If  we then conclude from the corollary of the Weak We must have m0  0 and m  0 . Duality Theorem that supL (m,n) : m  0 and We have that there exists a nonzero vector hence, (5) is satisfied. (m ,m ,n ) with (m ,m )  (0,0) such that 0 0 Thus, suppose that  is finite, and consider the following U U U system: (m0 ,m ,n ) [( p,q,r)  (0,0,0)]  m0 p  m q  n r  0 , for each cl . f (a* )   0, g(a* )  0,h(a* )  0 , for some aˆ*  A .

Also, note that[(a* ), g(a* ),h(a* )] cl and we have By the definition of  this system has no solution. Hence, from the above inequality that from the previous lemma, there exists a nonzero

* U * U * vector (m0 ,m,n) with (m0 ,m)  (0,0) such that m0(a )  m g(a )  n h(a )  0 . * U * U * * m0[ f (a )  ] m g(a )  n h(a )  0 , for all aˆ  A (6) Since the above inequality is true for each a*  A , we conclude that We will next show that m0  0 Suppose, by contradiction * U * U * System:1 m0(a )  m g(a )  n h(a )  0 for some that m0  0 * (m ,m,n)  (0,0,0),(m ,m)  (0,0) and for all a  A * 0 0 By assumption, there exists an aˆ*  A such that g(aˆ )  0 has a solution. and h(aˆ* )  0 . Substituting in (6) we obtain mU g(aˆ* )  0 .

To prove the converse, assume that m  0 . * U * 0 But since g(aˆ )  0 and m  0,m g(aˆ )  0is only possible * Suppose that a  A is such that g(a* )  0 . But since if m  0 . we must then have that(a* )  0 . From (6) and imply that nU h(a* )  0 for all aˆ*  A .but since 0int h(a* )we can choose an aˆ *  A System:2(a* )  0, g(a* )  0 , h(a* )  0 for some a*  A . such that h(a* )  n, where   0 . Has no solution and this completes the proof. Therefore0  nU h(a* )   v 2 which implies that n  0.

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Thus, it has been shown that implies * m0  0 Substitute f a   5 we get, 4  5 that (m ,m,n)  (0,0,0) which is a contradiction. We 0 Condition 2: m  0,ha*  0 & ga*  1 conclude, then, that m0  0 . Substitute the above condition in eqn. (8) we get, Dividing (6) by m and denoting 0 f a*  0  0  f a*  m  m m0 and n  n m0 we obtain f a*  f a*  * U * U * f (a )  m g(a )  n h(a )   for all aˆ*  A (7) Substitute f a*   5 we get, 5  5 To prove: Hence proved. ****UU L (,)infmn fa ()  mga ()  nhaa ():  A  Example 2: Strong duality We then conclude, from the Weak Duality Theorem, Strong duality equation is represented as, that (m,n)   and, from the corollary of the Weak Duality * U * U * f (a )  m g(a )  n h(a )   (9) Theorem, we conclude that (m,n) solves the dual problem. Given condition: f a*   , m u g a*  0 , n u h a*  0 Finally, to complete the proof, assume that a * is an optimal       solution to the primal problem; that u * u * Possible conditions: m ga 1,n ha 1 is, a *  A , g(a * )  0 , h(a * )  0 and f (a * )   . m u ga*  0,n u ha*  0 From (7), assuming a*  a * , we get u * u * U * * m g a 1,n h a  0 m g(a )  0 .since m  0 and g(a )  0 we     U * get m g(a )  0 m u ga*  0,n u ha* 1

This completes the proof. Condition 1: m u ga* 1,n u ha* 1, f a*  

Substitute the above condition in eqn. (9) we get, Example 1: Weak duality  11  Weak duality equation is represented as,

* u * u * *   2   L m,n f a  m ga  n ha  f a  (8) Substitute   5 5  2  5 Where, L m,nis the solution for dual problem and f a*  is the primal problem. 7  5 The value for dual problem must always be less than the value Condition 2: m u ga*  0,n u ha*  0 , f a*   of primal problem. Substitute the above condition in eqn. (9) we get, Given condition: ha*  0, m  0 and ga*  0 for all   0  0   ha*  0    Possible conditions: m 1& ga*  1; Substitute m  0 & ga*  1 5  5 * * Condition 1: m 1, g a  1& h a  0 u * u * *     Condition 3: m ga 1,n ha  0 , f a   Substitute the above condition in eqn. (8) we get, Substitute the above condition in eqn. (9) we get, * * f a 1(1)  0  f a   1 0   f a* 1 f a*   1  

3444 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3441-3448 © Research India Publications. http://www.ripublication.com

Substitute   5 Definition 1: Two kinds of concepts for no duality gap are presented below:

5 1 5 (i) We say that the primal problem (IVP) and the dual 6  5 problem (IVD) have no duality gap in the weak sense if and only if min f , A maxW ,Y   ∅ . Condition 4: m u ga*  0,n u ha* 1, f a*   (ii) We say that the primal problem (IVP) and the dual Substitute the above condition in eqn. (9) we get, problem (IVD) have no duality gap in the strong *   0 1  sense if and only if there exist f a min f , A and a* ,u* max ,Y such that  1   W   W * * * f a  W a ,u . Substitute Where, a* is a feasible solution to primal problem. a* ,u* is a

feasible solution to dual problem The fact that the primal problem (IVP) and the dual problem Hence proved. (IVD) have no duality gap in the strong sense implies that the primal problem (IVP) and the dual problem(IVD) have no * duality gap in the weak sense. We also see that, if a  a0 , FRANK-WOLFE DUALITY THEOREM then the fact that the primal problem (IVP) and the dual Strong and weak duality theorems are presented below. problem (IVD) have no dualitygap in the weak sense implies that the primal problem (IVP) and the dual problem(IVD) First of all, an ordering is introduced. Let A and B be two have no duality gap in the strong sense. sets of closed intervals. In conventional optimization problems, the weak duality theorem says that theobjective value of the primal problem is always greater than or equal to Theorem 2:Strong duality theorem in the weak sense the objectivevalue of the dual problem. In some sense, Let be an open subset of n . Let and min f , A and maxW ,Y  can be regardedas kinds of A0 R f optimal objective values of the primal problem (IVP) and dual Gx , x 1,2,.....k be convex and H-differentiable on . problem(IVD), respectively. Therefore, we are going to Suppose if one of the following conditions is satisfied, then present the weak duality theorem forproblems (IVP) and * (IVD) by showing that min f , Aand maxW ,Y  . (i) There exist a feasible solution a of the primal problem (IVP) such that Theorem 1: Weak duality theorem * f a objD W ,Y . n Let A0 be an open subset of R . Let f and (ii) There exist a feasible solution a* ,u* of the dual G , x 1,2,.....k be convex and H-differentiable x problem (IVD) such that on X Then, we have min f , A, max ,Y  . * * 0 W W a ,u objP  f , A . Proof: Then, the primal problem (IVP) and dual problem (IVD) have The result follows immediately from the equation below, no duality gap in the weak sense.

T T Proof The result follows immediately from the equation F a  W a,u below, In the sequel, we are going to present the strong duality  a ,u   f a*   a,u theorem by considering W 0 0 W Theorem 3: Strong duality theorem in the strong sense min f , A maxW ,Y   ∅ Let be an open subset of . Let and In other words, there exist f a* min f , A and be convex and H-differentiable on .  a ,u max ,Y  such that W 0 0 W Suppose that a* is a feasible solution of the primal problem * * * f a  W a0 ,u0  (IVP) and that (a ,u ) is a feasible solution of the dual problem (IVD) with the property that Therefore, we provide the following definition.

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k Example 3: u*G a*  0,0  x x     k x1 Given condition: * T * uxGx a  0 * * x1 Where u  u1,u2 ...... uk . Then a solves the primal * problem (IVP), (a* ,u* ) solves the dual problem (IVD) and Where, u  u the dual problem (IVD) has no duality gap in the strong sense. Possible conditions:

Condition: k * T * k uxGx a   1 * * T * * x1 uxGx a  0;Gx a  0;ux ,ux  0 x1 k * T * To prove: uxGx a   0 x1 k u G a*   0,0  x x Condition 1: x1 From the above equation we have that, Apply the above conditions in eqn. (10) we get, k * * * * * * k k  a ,u   f a  u G a   f a  T * * T * W  x x u G a  u G a x1  x x    x x   x1 x1 * * Let (a ,u ) be a feasible solution of the dual problem 1  1 * * m (IVD). Suppose that arg maxW a ,u , R  contains k * T * * Condition 2: u  0 and u infinitely large in magnitude.Then, a is a uxGx a   0 feasible solution of primal problem (IVP) and in the sequel, x1 we are going to provide sufficient conditions to guarantee that Apply the above conditions in eqn. (10) we get,

k k T * * T * uxGx a  uxGx a  x1 x1 k 0  0 * * * * * * W a ,u   f a  uxGx a   f a  x1 Hence proved. Proof: We have that a* ,u*  a* ,ufor all HYBRID LAGRANGE-WOLFE DUALITY THEOREM * * k Common Lagrange equation is represented as, u arg maxa ,u , R . j T T i By definition, we see that a* ,u*   a* ,u for all * * * * L m,n  inf{ f a  mx g x a  nx hx a  f a  . x1 x1 Sinceu,u*  0 , implies, Summation operation is removed in the above equation can be rewritten as, k k T * * T * T * T *  m* ,n*  f a*  m* g a*  n*h a*  f a* f a  uxGx a  f a  uxGx a  L           x1 x1 * The above equation can be written as, Where m  m

k k Common Wolfe equation is represented as, T * T * T * * T * f a  uxGx a  f a  uxGx a  k k x1 x1 * * T * * T * W a ,u   uxGx a  uxGx a  k k x1 x1 T * * T * uxGx a  uxGx a  (10) Summation operation is removed in the above equation and x1 x1 can be rewritten as,

3446 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3441-3448 © Research India Publications. http://www.ripublication.com

* * T * * T * W a ,u  uG a  u G a  Substitute Given conditions for lagrange theorem: 5  5 * * * ha  0;m,m 1,0, ga  1,0 Condition 4: k Given conditions for Wolfe m  1; g a*  1;h a*  0; u*GT a*  0 * T *      x x   theorem: ux 1,0;Gx a  1,0 x1 The equation for hybrid lagrange and wolfe can be Apply the above conditions in eqn. (11), we get represented as, f a* 11 0  0  f a*  0 ********* LW amnu,,,   fa   mga   nha  (11) * * kk f a 1 f a  TT**** ux G x a  f a  u x G x  a  xx11 Substitute Given condition: 4  5 Condition for both lagrange and wolfe duality theorem is represented below, Condition 5: k k m  0; g a*  1;h a*  0; u*GT a*  1 * * * * T *      x x   ha   0;m ,m  1,0; ga   1,0;uxGx a   0,1 x1 x1 Apply the above conditions in eqn. (11), we get Condition k * * * * * T * f a  01 0 1 f a 1 1: m  1; ga   1;ha   0;uxGx a   1 x1 f a* 1 f a* 1 Apply the above conditions in eqn. (11), we get Substitute f a* 11 0 1 f a* 1 6  6 f a*  f a* 1     Condition 6: * k Substitute f a   5 * * * T * m  0; ga   0;ha   0;uxGx a   1 x1 5  6 Apply the above conditions in eqn. (11), we get Condition 2: k * * * * * T * f a  0  0 1 f a 1

m  1; ga   0;ha   0;uxGx a   1 x1 Apply the above conditions in eqn. (11), we get Substitute f a* 10  0 1 f a* 1 6  6 f a* 1 f a* 1     Condition 7: * k Substitute f a   5 * * * T * m  0; ga   0;ha   0;uxGx a   0 x1 6  6 Apply the above conditions in eqn. (11), we get Condition 3: k * * * * * T * f a  0  0  0  f a  0 m  1; ga   0;ha   0;uxGx a   0 x1 f a*  f a*  Apply the above conditions in eqn. (11), we get Substitute f a* 10  0  0  f a*  0 5  5 f a*  f a* 

3447 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3441-3448 © Research India Publications. http://www.ripublication.com

Condition 8: [7] C.R. Bector, S. Chandra, “Generalized bonvex k functions and second order duality in Mathematical * * * T * m  0; ga   1;ha   0;uxGx a   0 Programming”, Department of Actuarial and x1 Management Sciences, The University of Manitoba, Winnipeg, January 1985. Apply the above conditions in eqn. (11), we get

* * [8] C.R. Bector, S. Chandra, “Second order symmetric f a  01 0  0  f a  0 and self-dual programs”, Opsearch, Vol. 23, no. 2, pp. 89–95, 1986. f a*  f a*  [9] Suneja SK, Lalitha CS, Khurana S. “Second order * symmetric duality in multi objective programming”, Substitute f a  5   European Journal of Operational Research, Vol. 4, 5  5 no. 3, pp.492-500, 2003. [10] Ahmad I, Husain Z. “On multiobjective second order Hence proved. symmetric duality with cone constraints”, European Journal of Operational Research, Vol. 204, no. 3, pp. 402-409, 2010. CONCLUSION [11] Bonnans JF, CominettiR, “Perturbed optimization in In this paper, nonlinear Lagrangian functions and Frank-wolfe Banach spaces a general theory based on a weak duality theorem was introduced for constrained multi- directional constraint qualification”, SIAM Journal objective optimization issues. This approach results in weak on Control and Optimization Vol. 34, pp.1151–1171, and strong duality and Sufficiency results based on both 1996. nonlinear Lagrangian functions and wolfe optimization. The constrained multi-objective optimization problem was [12] O.L. Mangasarian, “Second and higher order duality optimized using the obtained weak duality.This paper in nonlinear programming”, Journal of Mathematical proposed a theorem based on the combination of both Analysis and Applications, Vol. 51, pp.607–620, lagrangian and wolfe theorem. As a result, the proposed 1975. hybrid theorem solved multi-objective problems like [13] Kang, Y.M., Kim, D.S., Kim, M.H, “Optimality lagrangian theorem, convex, inequality, concave problems and conditions of G-type in locally Lipchitz multi invexity issues. objective programming”, Vol. 40, pp.275–285, 2012.

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