A Computationally Fast and Approximate Method for Karush-Kuhn-Tucker Proximity Measure

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1 A Computationally Fast and Approximate Method for Karush-Kuhn-Tucer Proximity Measure Kalyanmoy Deb and Mohamed Abouhawwash 1 Department of Electrical and Computer Engineering Computational Optimization and Innovation (COIN) Laboratory Michigan State University, East Lansing, MI 48824, USA deb@egr.msu.edu, mhawwash@msu.edu deb COIN Report Number Abstract Karush-Kuhn-Tucer (KKT) optimality conditions are used for checing whether a solution obtained by an optimization algorithm is truly an optimal solution or not by theoretical and applied optimization researchers. When a point is not the true optimum, a simple violation measure of KKT optimality conditions cannot indicate anything about it s proximity to the optimal solution. Past few studies by the first author and his collaborators suggested a KKT proximity measure (KKTPM) that is able to identify relative closeness of any point from the theoretical optimum point without actually nowing the exact location of the optimum point. In this paper, we suggest several computationally fast methods for computing an approximate KKTPM value, so that a convergence measure for iteration-wise best solutions of an optimization algorithm can be quantified for terminating an optimization run. The KKTPM value can also be used to isolate less-converged solutions in a population-based optimization algorithm so as to specially modify them for an overall faster execution of the optimization run. The approximate KKTPM values are evaluated in comparison with the original exact KKTPM value on standard single-objective, multi-objective and many-objective optimization problems. In all cases, our proposed estimated approximate method is found to achieve a strong correlation of KKTPM values with the exact values and achieve such results in two or three orders of magnitude smaller computational time. These results are extremely motivating to launch further studies in using the proposed estimated KKTPM procedure for establishing termination-based and developing other modified optimization procedures. Keywords: Multi-objective optimization, Evolutionary optimization, KKT optimality conditions, method, method. 1 Also Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. saleh1284@mans.edu.eg.

2 1. Introduction Karush-Kuhn-Tucer (KKT) optimality conditions [1, 2] are necessary requirement for an optimal solution to a single or multi-objective optimization problem to satisfy [3, 4, 5]. These conditions require first-order derivatives of objective function(s) and constraint functions, although extensions of them for handling non-smooth problems using subdifferentials exist [6, 4]. Although KKT conditions are guaranteed to be satisfied at optimal solutions, the mathematical optimization literature is mostly silent about the regularity in violation of these conditions in the vicinity of optimal solutions. This lac of literature prohibits optimization algorithmists woring particularly with approximation based algorithms that attempt to find a near-optimal solution at best to chec or evaluate the closeness of a solution to the theoretical optimal solution using KKT optimality conditions. However, recent studies have suggested interesting new definitions of neighboring and approximate KKT points [7, 8, 9], which have been resulted in a KKT proximity measure [1, 7]. The study has clearly shown that a naive measure of the extent of violation of KKT optimality conditions cannot provide a proximity measure to the theoretical optimum, however, the KKT proximity measure (KKTPM) that uses the approximate KKT point definition maes a measure of closeness of a solution from the KKT point. This is a remarable achievement in its own right, as KKTPM allows a way to now the proximity of a solution from a KKT point without actually nowing the location of the KKT point. A recent study has extended the KKTPM procedure developed for single-objective optimization problems to be applied to multi-objective optimization problems [11]. Since in a multi-objective optimization problem with conflicting objectives, multiple Pareto-optimal solutions exist, KKTPM procedure can be applied to each obtained solution to obtain a proximity measure of the solution from its nearest Pareto-optimal solution. The KKTPM computation opens up several new and promising avenues for further development of evolutionary single [12] and multi-objective optimization algorithms [13, 14]: (i) KK- TPM can be used for a reliable convergence measure for terminating a run, (ii) KKTPM value can be used to identify poorly converged non-dominated solutions in a multi-objective optimization problem, (iii) KKTPM-based local search procedure can be invoed in an algorithm to speed up the convergence procedure, and (iv) dynamics of KKTPM variation from start to end of an optimization run can provide useful information about multi-modality and other attractors in the search space. However, since the exact KKTPM computation involves a nested optimization tas in an algorithm, the computational effort is a bottlenec for the above development. In this paper, we analyze the KKTPM computation procedure and suggest three approximate algorithmic procedures for a fast computation of KKTPM. For certain points including the KKT point or optimal solution, the proposed approximate KKTPM values are identical to the exact KKTPM value and we are able to find a implementable condition when this happens. In other cases, we find two approximate methods that are guaranteed to bracet the exact KKTPM value and another approximate method to obtain a closer to exact KKTPM value. We then demonstrate the use of approximate KKTPM methods and show the accuracy of the proposed estimated KK- TPM values compared to the exact KKTPM values and importantly report two or three orders of magnitude faster computation with the approximate method. In the remainder of the paper, we first discuss the importance for computing KKTPM in Section 2. Then, in Section 3, we describe the principle of KKT proximity measure for single and multi-objective optimization problems based on the achievement scalarizing function (ASF) concept proposed in the multiple criterion decision maing (MCDM) literature. In Section 4, we propose three fast yet approximate methods for computing the KKT proximity measure without 2

3 using any explicit optimization procedure. Section 5 suggests to replace ASF with its augmented version so as to avoid wea Pareto-optimal solutions. Section 6 compares all three proposed approximate methods with previously-published exact KKTPM values on standard single, multiand many-objective optimization problems including a few engineering design problems. Finally, conclusions are made in Section Need for Computing The earlier study by the authors [11] have clearly demonstrated that the recently proposed KKT proximity measure (KKTPM) provide a quantitative measure of proximity of populationbest solutions in an evolutionary algorithm to theoretical optimal solutions, which then can be used as a termination condition. Although the study did not simulate a classical point-by-point optimization method, the proposed KKTPM is also applicable for classical optimization methods. A recent study [15] demonstrated that evolutionary multi-objective optimization (EMO) methods constitute a parallel search in finding multiple trade-off solutions in a computationally quicer manner compared to their point-by-point generative approaches [16, 17]. Such exploratory and application studies are maing EMO methods popular and useful in practice. For solving single-objective optimization problems, the best solution at every generation (or iteration) can be recorded and its KKTPM value can be computed. As demonstrated in the original study [7, 11], the KKTPM value reduces almost monotonically to zero as the iterate approaches the optimal solution (or KKT point). The study has also shown the correlation between the distance from the nown optimal solution and the corresponding KKTPM value. Thus, a threshold on the KKTPM value can be enforced for a suitable termination of an optimization run. This will avoid any guesswor for choosing a pre-specified number of generations or function evaluations, or stability of objective value of current best solution over the past few generations. Besides allowing an automatic termination, it also provides confidence in the final solution, as a small value of KKTPM guarantees the solution to be close to a KKT point. For multi- and many-objective optimization problems, the non-dominated solutions at every generation of an EMO can be recorded and their KKTPM values can be computed. Although these solutions are non-dominated to each other, their closeness to the true Pareto-optimal front are liely to be different. Thus, the respective KKTPM value for each non-dominated solution at a generation is expected to be different from each other. A few statistics of these KKTPM values (such as smallest, median and largest) can be computed and analyzed for a termination. Although the smallest and largest KKTPM values may not be good indicators of convergence of the entire set, the median KKTPM value can be checed against a threshold value. In such a case, a termination will then mean that at least 5% of the trade-off non-dominated solutions have come to close to the Pareto-optimal front with the specified threshold value. There exist a few other reasons for pursuing to compute KKTPM, which we highlight here. 1. It is important to realize that not all non-dominated solutions are liely to be close to the Pareto-optimal front. Figure 1 shows that with respect to a few true Pareto-optimal solutions (with KKTPM value equal to zero for each of them) shown in filled circles, two types of solutions can be non-dominated. First, there can be solutions that are close to the Pareto-optimal front, but are not truly Pareto-optimal. These points are shown in open circles. Second, there can be solutions that are not even close to the Pareto-optimal front, but are non-dominated to the rest of the trade-off solutions. These solutions are mared in open boxes. A computation of KKTPM for each of these solutions will reveal the 3

4 f2 Small KKTPM Zero KKTPM Large KKTPM Pareto optimal front f1 Figure 1: A set of non-dominated points may have widely different KKT proximity measure values. properties of each of the three types of non-dominated solutions. For true Pareto-optimal (in the sense of theory) solutions, KKTPM value will be guaranteed to be zero. For near Pareto-optimal solutions, KKTPM value will be small and for far away solutions, KKTPM value will be large. For the first time, the above observation provides an unique information about individual non-dominated population members to an EMO algorithm. So far, the only way to differentiate non-dominated solutions in an EMO procedure is based on their neighborhood information [18, 19] or closeness to a reference direction [2, 21]. For the first time, KKTPM computation opens up a unique way to differentiate different non-dominated solutions in an EMO population based on their relative proximity to true Pareto-optimal front. Special action can then be taen to fix the non-dominated solutions that are away from the true Pareto-optimal front to obtain a faster overall convergence. We do not address these additional and special efforts in this paper, but before we are able to do them, we need a faster way to compute KKTPM value, which is the main goal of this paper. 2. In many problems, objective values near optimal or Pareto-optimal region can tae more or less similar values (maing the landscape somewhat flat ). In such problems, evolutionary optimization algorithms lac an adequate search power to converge close to the true optimal or Pareto-optimal solutions quicly, rather populations wander around them with elapse of generations. For problem SRN, Figure 2 shows the NSGA-II population after 25 generations. While the true Pareto-optimal solutions are along the vertical line at x 1 = 2.5, NSGA-II solutions wander around the optimal line until 5 generations. Since evolutionary optimization methods do not use any gradient information, there is no real way to salvage any second-order information to attain better convergence. Since it has been established that KKTPM has a direct correlation with the distance of a point from its nearest optimum, it can be provide the additional information needed to distinguish a relatively close solution to an optimum than one that is away. Thereafter, the use of an appropriate local search method can be employed to mae the solution converge to the optimal solution. Again, we do not address this aspect of fine-tuning near optimal solutions 4

5 x x1 Figure 2: NSGA-II population members wander around Pareto-optimal solutions ever after 25 generations. in this paper, but we are currently pursuing a separate study and shall report results at a later publication. It is amply clear from the above discussions that besides using KKTPM to aid an automatic and more confident termination of an optimization run, KKTPM can be used to expedite the search process better. However, there are a few bottlenecs for computing and using KKTPM which we discuss next. 1. The original study [11] proposed an optimization procedure for computing the exact KK- TPM value. This optimization procedure uses Lagrange multipliers (one for each constraint or each variable bound) and two additional parameters (KKTPM itself and a slac variable for converting a non-differentiable problem into a smooth problem) as variables. The problem also has two main inequality constraints one quadratic and one linear to Lagrange multipliers. The solution of this optimization problem can be time-consuming, particularly if KKTPM value needs to be computed for every population member in each generation. In this paper, we address this issue and avoid solving the optimization problem by suggesting a computationally fast and approximate method without sacrificing much on the accuracy of KKTPM value. With this ability, we believe that KKTPM approach paves the way of its use in the above-mentioned algorithmic modifications for a faster overall approach. 2. Second, KKPTM requires gradient information of the original objective functions and constraints at the point at which the KKTPM value needs to be computed. The original study used exact gradients for most examples but demonstrated by woring with numerical gradient computation (forward difference method) on a single problem that the induced error is small and is proportional to the stepsize used for numerical gradient computation. This is encouraging and we plan to pursue this aspect in more detail at a later study. Other interval-based KKT optimality conditions [22] will also be explored. 3. Third, a zero KKTPM value can come from both local and global optimal solutions. However, this is not a big issue, as in addition to KKTPM value, a solution s objective value 5

6 can be used to differentiate local and global solutions, but a hierarchical selection operator with KKTPM value followed by the objective value (or non-domination level) can be used to steer the search towards the global optimal solution. 4. Fourth, KKTPM value indicates a measure of a solution s proximity to the respective optimal solution. In multi-objective optimization, in addition to convergence, diversity preservation among non-dominated solutions is another equally important goal. Thus, a termination as well as other modifications suggested above must accompany a careful consideration between convergence and diversity of solutions. Fortunately, the convergence issue can now be dealt with KKTPM reliably and further research is needed to combine it with a diversity measure for achieving a fast and reliable optimization. Setting the stage for highlighting the importance and role of KKTPM in enhancing performance and reliably terminating an optimization algorithm from a convergence issue, we now return to the main goal of this paper a fast and approximate computation of KKT proximity measure. We highlight the fact that if we are able to achieve a fast computation of KKTPM without sacrificing much on the accuracy, it will then open up a number of avenues for algorithmic improvements, as discussed above. In the following, we first present a review of the philosophy behind computing KKTPM and then suggest the proposed fast but approximate procedure. 3. Review of KKT proximity measure was first proposed elsewhere [7] for single-objective optimization problems. Since KKT error measure, derived from the violation of KKT equilibrium conditions, has a singularity property at any KKT point and is unable to relate the proximity of a solution to a KKT point, the development of KKT proximity measure (KKTPM) was a seminal achievement and for the first time it demonstrated its importance in arriving at a suitable termination condition. Recently, the single-objective KKT proximity measure has been extended for multi-objective optimization problems [11]. In this section, we mae a brief introduction to both studies, so that we can then build up a fast computational method in the next section for Single-Objective Optimization Dutta et al. [7] defined an approximate KKT solution to compute a KKT proximity measure for any iterate x for a single-objective optimization problem of the following type: Minimize (x) f (x), Subject to g j (x), j = 1, 2,...,J. (1) After a long theoretical calculation, they suggested a procedure of computing the KKT proximity measure for an iterate (x ) as follows: Minimize (ɛ,u) ɛ Subject to f (x ) + J j=1 u j g j (x ) 2 ɛ, J j=1 u jg j (x ) ɛ, u j, j, (2) However, that study restricted the computation of the KKT proximity measure for feasible solutions only. For an infeasible iterate, they simply ignored the computation of the KKT proximity 6

7 measure. For the above optimization problem, the variables are (ɛ, u). The KKT proximity measure (KKTPM) was then defined as follows: (x ) = ɛ, (3) where ɛ is the optimal objective value of the problem stated in Equation 2. To illustrate the woring of KKTPM, we consider a two-variable constrained minimization problem: Minimize f (x 1, x 2 ) = x x2 2, s.t. g 1 (x 1, x 2 ) 3x 1 x 2 + 1, g 2 (x 1, x 2 ) x (x 2 2) 2 1. (4) Figure 3 shows the feasible space and the optimum point (C) (x = (, 1) T. For points along line 3.5 x O B C g 2 f D g1= A f g 1 f g 1 g 2 g 2 g2= x1 Figure 3: Feasible search space of the example problem. g 1 KKTPM Value A Distance from A Along AC Figure 4: Variation of KKTPM value along line AC from point A for the example problem. C AC, Figure 4 shows the variation of KKTPM. It is clear that as points get closer to the optimal point, KKTPM value reduces monotonically and finally becomes zero at the optimal point Exact for Multi-Objective Optimization The theory for the development of the exact KKTPM for multi-objective optimization problems was presented in an earlier study by the authors [11]. Here, we summarize the main results for completeness. For a n-variable, M-objective optimization problem with J inequality constraints: Minimize (x) { f 1 (x), f 2 (x),..., f M (x)}, Subject to g j (x), j = 1, 2,...,J, (5) the Karush-Kuhn-Tucer (KKT) optimality conditions for Equation 5 are given as follows [6, 7

8 17]: M u m f m (x ) + m=1 J u j g j (x ) =, (6) j=1 g j (x ), j = 1, 2,...,J, (7) u j g j (x ) =, j = 1, 2,...,J, (8) u j, j = 1, 2,...,J, (9) u m, m = 1, 2,...,M, and u. (1) Multipliers u m are non-negative, but at least one of them must be non-zero. The parameter u j is called the Lagrange multiplier for the j-th inequality constraint and they are also non-negative. Any solution x that satisfies all the above conditions is called a KKT point [4]. Variable bounds of the type x (L) i x i x (U) i andg J+2i (x) = x i x (U) i can be splitted into two inequality constraints: g J+2i 1 (x) = x (L) i x i. Thus, in the presence of all n pairs of specified variable bounds, there are a total of J + 2n inequality constraints for the above problem. For a given iterate (solution), x, the original study [11] formulated an achievement scalarization function (ASF) optimization problem [23]: ( Minimize (x) ASF(x, z, w) = maxm=1 M fm ) (x) z m w m, Subject to g j (x), j = 1, 2,...,J. (11) The reference point z R M was considered as an utopian point and the weight vector w R M is computed for x as follows: f i (x ) z i w i =. (12) M m=1 ( f m(x ) z m ) 2 Thereafter, the KKT proximity measure was computed by applying the KKTPM computation procedure developed for single-objective optimization problems stated in equation 2 to the ASF formulation given above. Since the ASF formulation maes the objective function non-differentiable, a smooth transformation of the ASF problem was first made by introducing a slac variables x n+1 and reformulating the original problem as follows: Minimize ( F(x, x n+1 ) ) = x n+1, Subject to fi (x) z i x w n+1, i = 1, 2,...,M, i g j (x), j = 1, 2,...,J. (13) Now, the KKTPM optimization problem for the above smooth single-objective problem for y = (x; x n+1 ) can be written as follows: Minimize (ɛ,x n+1,u) ɛ + ( J j=1 um+ j g j (x ) ) 2, Subject to F(y) + M+J j=1 u j G j (y) 2 ɛ, M+J ( j=1 u jg ) j (y) ɛ, f j (x) z j x w n+1, j = 1,...,M, j u j, j = 1, 2,...,(M + J). 8 (14)

9 The added term in the objective function allows a penalty associated with the violation of complementary slacness condition [11]. The constraints G j (y)aregivenbelow: f j (x) z j G j (y) = x n+1, j = 1,...,M, (15) w j G M+ j (y) = g j (x), j = 1, 2,...,J. (16) The optimal objective value (ɛ ) to the above problem corresponds to the exact KKT proximity measure. It is observed that for feasible solutions ɛ 1, hence the exact KKTPM was defined as follows: ɛ Exact (x ) =, if x is feasible, 1 + J j=1 g j (x ) 2 (17), otherwise. 4. Proposed Fast and Approximate Computation of KKTPM The above exact KKTPM computation procedure requires an optimization problem (equation 14) to be solved for every iterate x. Since the variables for this optimization tas are (ɛ, x n+1, u) (M + J + 2) of them, the computational time to compute KKTPM value for every non-dominated solution at every generation of an EMO can be large. To mae a fast computation of an approximation of the KKTPM value, we suggest a few fast but approximate methods here KKTPM Computation Method There are two main constraints to the optimization tas in equation 14. The first constraint requires the computation of the gradient of F and G functions and can be written as: M ɛ u j w j=1 j 2 2 J f j (x ) + u M+ j g j (x M ) + 1 u j. (18) The second constraint in equation 14 can be written as follows: ɛ j=1 M f j (x u j ) z j J x n+1 u M+ j g j (x ). (19) j=1 w j Since x n+1 is a slac variable, the third set of constraints will get satisfied by setting x n+1 = M f j (x ) z j. (2) max j=1 For any feasible solution x, f j (x ) is always greater than the respective j-th component of any utopian point z. Hence, x n+1 is also non-negative. Thus, the constraint set of the problem given in equation 14 reduces to the first and second constraints and the non-negativity constraint of the Lagrange multipliers u = (u M, u J ), where u M and u J are M-dimensional and J-dimensional vectors of Lagrange multipliers. w j j=1 j=1 9

10 Let us consider the first constraint alone. It is clear that it is a quadratic function in terms of Lagrange multipliers u and can be written as follows: n M+J 2 2 M ɛ u i a ij + 1 u i, (21) where a ij can be written in terms of derivatives of f j and g j functions: a ij = 1 f i w i j=1 i=1 i=1 x x j, for i = 1, 2,...,M, g (i M) x x j, for i = M + 1, M + 2,...,M + J, for j = 1, 2,...,n. Writing the above constraint in terms of vectors u M and u J,wehave ɛ ( A T M u M + A T J u J) T ( A T M u M + A T J u J) + (1 1 T M u M ) 2, (22) where 1 M is a M 1-dimensional matrix of ones, A M is (M n)-dimensional matrix and A J is (J n)-dimensional matrix, extracted from the overall ((M + J) n)-dimensional matrix [ ] AM A = [a ij ] =. In our first approximation (we refer here as the ) method, we assume that the second constraint given in equation 19 is inactive and the first constraint is active at the optimal solution to the KKTPM optimization problem given in equation 14. Thus, the problem becomes as follows: A J Min. (ɛ,u M,u J ) ɛ + u T J GGT u J, Subject to ɛ ( A T M u M + A T J u T ( J) A T M u M + A T J u J) + (1 1 T M 1 u M ) 2, u M, u J, (23) where G is the J-dimensional constraint vector (g 1 (x ), g 2 (x K ),...,g J (x ) T.Sinceɛ K is an independent variable, the above optimization problem can be simplified as follows: Min. (um,u J ) ( A T M u M + A T J u ) T ( J A T M u M + A T J u ) J +(1 1 T M 1 u M ) 2 +u T J GGT u J, Subject to u M, u J, (24) The above objective function is a quadratic function in terms of u M and u J.Bydifferentiating the objective function with respect to u M and u J,wehave A M A T M u M + A M A T J u J ( 1 1 T M 1 u M) 1M 1 = M 1, (25) A J A T M u M + A J A T J u J + GG T u J = J 1. (26) Rearranging the above two vector equations, we have [ AM A T M + 1 ]( ) M M A M A T J um A J A T M A J A T J + GGT u J 1 = ( ) 1M 1. (27) J 1

11 Interestingly, the above equations are a set of linear equations, which can be solved to obtain (ū M, ū J ) T vector. However, it is not necessary that each component of this vector be non-negative. To satisfy the non-negativity of constraints, if any ū M or ū J is calculated to be negative, we set them to zero, and eliminate that equation from above set of linear equation and re-solve the remaining equations. We continue this process until the non-negativity of u M and u J are established. Let us say that the final vector is (u D M, ud J )T. By transposing and post-multiplying equation 25 by u D M and doing the same for equation 26 by u D J, we obtain (A T M ud M + AT J ud J )T A T M ud M = ( 1 1 T M 1 M) ud 1M 1 u D M, (28) (A T M ud M + AT J ud J )T A T J ud J = ( ) u D T J GG T u D J. (29) Substituting these left-hand terms for ɛ term and manipulating, we have the direct approximated KKTPM value arising only from the first (quadratic) constraint: ɛ D = ( 1 1 T M 1 M) ud 1M 1 u D M ( ) u D T J GG T u D J + (1 1T M 1 ud M )2, = 1 1 T M 1 ud M ( ) G T u D 2 J. (3) Since um D, u D j, and GG T is a matrix with positive elements, it can be concluded that ɛ D 1 for any feasible iterate x. This result also supports our earlier definition of the exact KKTPM value for infeasible solutions, as described in equation 17. The above-computed ɛ D value corresponds to the exact ɛ value, if at the above (ud M, ud J )T vector, the second constraint is also satisfied. Using equations 12 and 13, we obtain the following condition for this to be true: M x n+1 ( f (x ) z ) 2. (31) =1 Since x n+1 is an independent variable, the minimization of objective function at equation 14 will force x n+1 to tae the smallest positive value. Hence, the equation will be satisfied with equality sign. This will simplify equation 19 as follows: ɛ G T u J. (32) Therefore, ɛ D = ɛ happens only when the following condition is true at (ud M, ud J )T vector: or, 1 1 T M 1 ud M ( ) G T u D 2 J G T u J, (33) 1 T M 1 ud M ( )( ) G T u D J 1 G T u D J 1. (34) Here is the summary of our approach so far. After finding the feasible solution of the linear system of equation 27 with non-negativity restriction of u D -vector, it can be used to chec the above condition. If the condition is satisfied, ɛ D is identical to the exact KKTPM value (ɛ ). If this happens, we are then able to avoid a complete optimization run to compute ɛ and have managed to find a computationally faster method of computing it. This scenario is illustrated with a setch in Figure 5 with a single Lagrange multiplier u j. The first constraint is quadratic to u j (equation 18) and the second constraint is a linear function of u j (equation 32). In a generic case 11

12 ε First constraint Feasible region B A ε Feasible region ε =ε opt D Second constraint u ε ε ε ε Adj P opt D A P O D Second constraint First constraint u Figure 5: First constraint governs at iterate x which is not a KKT point (A) and a KKT point (B). Figure 6: Second constraint governs at iterate x which is not a KKT point. satisfying condition 34, the second constraint becomes redundant in the KKTPM optimization process having ɛ and the use of the first constraint is adequate to find the optimal ɛ.also, due to the solution of a set of linear equations, the computation of ɛ becomes faster. An interesting scenario to consider is the case when the iterate x is a KKT point (or an optimal point). In this case, the second constraint passes through the minimal point of the first constraint function and condition 34 becomes 1 T M 1 ud M ( )( ) G T u D J 1 G T u D J = 1. (35) At the KKT or an optimal point, the complementary slacness condition is satisfied, maing G T u D J =. With this condition, equations 26 and 25 yields 1T M 1 ud M = 1. These two conditions satisfy the above identity, thereby meaning that at the optimal or a KKT point, the above identity (equation 35) is always true. The above brings out an interesting property of a KKT or an optimal point using our derivation in this paper the respective Lagrange multipliers for a KKT or an optimal point can be scaled so that the sum of Lagrange multipliers u m for objective functions is one. Although original KKT conditions for an optimal or KKT point x may not mae M =1 u equal to one, our formulation for KKT proximity measure forces this condition to be true. This can be explained as follows. Let us consider the KKT optimality conditions given in equations 6 to 1 again. By dividing these equations by a non-negative quantity M =1 u and writing KKT conditions for the normalized Lagrange multipliers ū = u / M =1 u and ū j = u j / M =1 u, we have the following 12

13 modified KKT conditions: M J ū m f m (x ) + ū j g j (x ) =, (36) m=1 j=1 g j (x ), j = 1, 2,...,J, (37) ū j g j (x ) =, j = 1, 2,...,J, (38) ū j, j = 1, 2,...,J, (39) ū m, m = 1, 2,...,M, and ū. (4) Since a KKT or an optimal point x must also satisfy the above conditions, the sum of modified Lagrange multipliers ū m can be written as follows: M ū m = m=1 M m=1 u m M=1 u = 1. (41) Let us now get bac to a more generic scenario in which the condition 34 is not satisfied for an iterate x. In this scenario, ɛ ɛ D and in fact, ɛ >ɛ D. The scenario is illustrated in Figure 6. The second constraint (line) now dictates the location of the optimal ɛ,whichis mared as the point O in the figure. The feasible region due to both constraints are shown shaded. The minimum ɛ occurs at the point O. The only way to find this point exactly is to solve the optimization problem given in equation 14, which can be computationally demanding. Note that for a single u j, the intersection of the quadratic and linear constraint functions occurs at only two points, as shown in the figure, and choosing the intersection point with smaller ɛ value will ensure finding ɛ. However, for more than one u j elements, the intersection between the quadratic and linear constraint functions produces an ellipsoidal intersecting surface having infinite possibilities. The use of an minimization method to choose the minimum ɛ value of all intersection points will produce ɛ ; hence the need for using an optimization procedure. Note that this optimum ɛ occurs at the intersection of a quadratic function and a linear function. Thus, a quadratic programming (QP) method cannot be used, rather a sequential quadratic programming (SQP) method must be used. This is why in the original study [11], we have used Matlab s fmincon routine, in which SQP is an option. There is no other computationally simpler way to avoid the optimization run to compute ɛ for this scenario. However, we can find approximate values of ɛ by using computationally faster approaches, which we discuss in the next few subsections KKTPM Computation Method From the direct solution u D and corresponding ɛ D (point D in the Figure 6), we compute an adjusted point A (mared in the figure), by simply computing the ɛ value from the second constraint boundary at u = u D, as follows: ɛ Adj = G T u D J. (42) From the geometry, it is easy to realize that this adjusted KKTPM value, ɛ Adj, will always overestimate the optimal ɛ. Moreover, since ɛd is the minimum of the quadratic constraint, the following relationship is always true: ɛ D ɛ ɛadj. (43) 13

14 4.3. KKTPM Computation Method Next, we consider another approximation method using u D j. This time, we mae a projection from the direct solution (point D ) (u D,ɛ D ) on the second constraint boundary. We then obtain the projected point P on the second constraint. Let us consider that the projected point has a coordinate (u P,ɛ P). The linear (second) constraint boundary can be written as ɛp + GT u P J =. The perpendicular distance, d PD, from point D to the linear constraint boundary is given as d PD = ɛd + GT u D J 1 + GT G. The unit vector along line PD is ( 1, G T ) T. Writing PD in two different ways and equating them, we have ( ) ɛ P ɛ D u P J ud J ( ) 1 = d PD G = ɛd + GT u D J 1 + G T G ( ) 1. G From the first element of the above vector equation, we estimate the projected KKTPM value, as follows: ɛ P = ɛ D ɛd + GT u D J 1 + G T G, ) ( = GT ɛ DG ud j 1 + G T G. (44) While ɛ P is always smaller than ɛadj, ɛ P can be larger or smaller than ɛ depending on the shape and location of the two constraints. The projected KKTPM value can either be smaller or larger than the exact KKTPM value, but is always guaranteed to be within ɛ D and ɛadj values KKTPM Computation Method Due to uncertainty in the range of the braceting ɛ values (ɛ D and ɛ Adj ) for the optimal ɛ and due to the fact that ɛ P always lie within these two bounds, we suggest taing an average of the three approximated ɛ values as our proposed estimated KKTPM value: ɛ est = 1 3 ( ɛ D + ɛ P + ) ɛadj. (45) Due to above properties, this measure can be regarded as a more reliable approximate to the exact KKTPM value than the other three approximate KKTPM values. It is intuitive to realize from Figure 5 that for a KKT point or the optimal iterate x,the following condition is always true: ɛ D = ɛadj = ɛ P = ɛest = ɛ =. (46) 14

15 5. with Augmented ASF The ASF procedure calculated from an ideal or an utopian point may result in a wea Paretooptimal solution [13]. To avoid, an augmented version of ASF was proposed [17]: ( fi (x) z i Minimize (x) ASF(x, z, w) = maxi=1 M w i Subject to g j (x), j = 1, 2,...,J. ) + ρ Mi=1 ( fi (x) z i w i ), (47) Here, the parameter ρ taes a small positive value ( 1 4 ). The approximate KKT proximity metric computation procedures (ɛ D from equation 3, ɛadj from equation 42, ɛ P from equation 44, and ɛ est from equation 45) can be easily extended using the above augmented ASF formulation. For brevity, we do not show the calculations here. For all simulations of this paper, we have used the augmented ASF function so that wea Pareto-optimal solution will have a non-zero KKTPM value. 6. Results In this section, we consider single objective, multi-objective and many-objective test problems to demonstrate the woring of the proposed approximate KKT proximity measures for solutions obtained using different evolutionary optimization algorithms. Single-objective problems are solved using the elite-preserving real-coded genetic algorithm [24] with p c =.9andη c = and polynomial mutation operator [13] with p m = 1/n (where n is the number of variables) and η m = 2. A C-programming code is available from website For bi-objective problems, we use NSGA-II [18] and for three or many-objective problems we use the recently proposed NSGA-III procedure [21, 25]. For problems solved using NSGA-II and NSGA-III, we use the simulated binary crossover (SBX) operator [24] with p c =.9 and η c = 3, and polynomial mutation operator [13] with p m = 1/n (where n is the number of variables) and η m = Single-Objective Optimization Problems We start with results on a simple problem and then present results on standard G-series of constrained test-problems [26] A Proof-of-Principle Example Problem First, we consider a single-variable, single-constraint problem to mae different approximate methods proposed in this study clear: Minimize f (x) = x 2, Subject to g(x) =.5 x. (48) The optimal solution is x =.5 and by theory, ɛ = at this solution. Noting f (x) = 2x and g(x) = 1, the first constraint (18) at any iterate x () becomes: ɛ u 1 f (x () ) + u 2 g(x () ) 2 + (1 u 1 ) 2, or, ɛ (2u 1 x () u 2 ) 2 + (1 u 1 ) 2. The second constraint at iterate x becomes ɛ u 2 (.5 x () ). (49) 15

16 Let us now tae two x () values and investigate the difference between exact and approximated KKTPM values. For x () =.5 (the optimal solution to the above optimization problem), two constraint surfaces are given as follows: ɛ (u 1 u 2 ) 2 + (1 u 1 ) 2, (5) ɛ, (51) Figure 7 shows these two constraint surfaces. The z-axis indicates the ɛ. It is clear that the first Eps x=.5 2 1st 1.5 constr nd constr u u2.8 1 Eps u2 ε P ε opt 1st constraint ε Adj 2nd constr. ε D u1 3 2 Figure 7: Two constraint surfaces are shown for x () =.5 (optimal solution). Figure 8: Two constraint surfaces are shown for x () = 1. (an non-optimal feasible solution). constraint value is never smaller than the second constraint value at any point (u 1, u 2 ) T. Hence, we have a scenario similar to that discussed in Figure 5. Interestingly, the two constraint surfaces meet at (u m, u j ) = (1, 1) T and the corresponding ɛ =. First, we use the direct method and formulate the optimization problem given in equation 24 using the first constraint, as follows: Min. (u1,u 2 ) (u 1 u 2 ) 2 + (1 u 1 ) 2, Subject to u 1, u 2. (52) The solution to the above problem is u1 D = ud 2 = 1 with ɛd =. Thus, the direct method maes an accurate estimation of the optimal ɛ. It is also interesting to note that since G = (.5.5) = for this solution, equations 42 and 44 estimate ɛ Adj = ɛ P = andɛest =, thereby supporting our conclusion about exact and approximated KKTPM values (equation 46) for optimal solutions. Next, we consider a feasible but non-optimal solution x () = 1. At this point, the solution of the KKT optimality conditions yields in the optimal KKTPM value as ɛ = Let us now estimate different approximate KKTPM values. First, we compute ɛ D using the direct method. For this purpose, we formulate the following quadratic problem: Min. (u1,u 2 ) ɛ = (2u 1 (1) u 2 ) 2 + (1 u 1 ) 2 + u 2 2 (.5 1)2, Subject to u 1, u 2. (53) The solution to this problem is u1 D =.556, ud 2 =.889 and corresponding ɛd =.2469, which is smaller than the exact KKTPM value (ɛ =.3441). Figure 8 mars the quadratic constraint 16

17 function and corresponding ɛ D value. Next, we observe that the second constraint function is as follows: ɛ.5u 2. (54) Figure 8 shows that ɛ D surface lies below this linear constraint surface at ud j. Putting ud m and u D j values found above to condition 34, we find that left side value is 1.197, which is greater than 1, thereby violating the condition. Since condition 34 is violated, ɛ D ɛ. Using equations 42, 44 and 45, we then obtain the following estimations of KKTPM: ɛ Adj =.4444, ɛ P =.449, ɛest = Note that exact KKTPM (.3441) is bounded within direct KKTPM (ɛ D =.2469) and adjusted KKTPM (ɛ Adj =.4444). Moreover, due to the use of ɛ P in the averaging process, the estimated KKTPM =.3654 is closer to the exact KKTPM value of.3441 than the two braceting KK- TPM values. This example clearly illustrates the relationship of each approximated KKTPM value with the exact KKTPM value G-Series Constrained Problems Now, we are ready to present results of our proposed approximation methods comparing with exact KKTPM values on standard single-objective constrained optimization problems g- functions [26]. We apply an elitist RGA [24] and record the population-best solution x () at every generation. Thereafter, we compute exact and approximate KKTPM values and compare. Figure 9 is for g1 problem from [26] having 35 constraints and 13 variables. We run an elite-preserving real-coded genetic algorithm (RGA) [27, 28] for 2 generations and with 1 population size. The KKT proximity measure of the population-best solution is calculated using optimal (shown in solid line), direct, projection, adjusted, estimated (shown with upside down triangles) methods, discussed above. The proposed estimated values are close to the exact KK- TPM value. It can also be observed that the ɛ is always bounded by two approximated KKTPM values: ɛ D (pin line) and ɛ Adj (green line). For this problem, RGA is able to find the true optimum (1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1) T with f = at generation 94. The exact KKTPM value ɛ is found to be exactly zero. The inset figure shows this fact in a semi-log plot better. At this generation, all approximated KKTPM values are also zero. The population-best solution for the above run is infeasible until generation 1, then in generation 11, the first feasible solution is found. Notice how all KKTPM measures including approximated ones have identical and large value (more then one) until generation 1 and thereafter when feasible solutions are found, the approximated KKTPM values are different from the exact KKTPM value. But the estimated KKTPM matches well with the exact KKTPM value. Importantly, the relative change in exact KKTPM values over generations is in agreement with changes in the estimated KKTPM values. We compute the correlation coefficient between ɛ and ɛest l and observe the value is.9897, meaning that there is very strong positive correlation in the changes in two sets of KKTPM values. Next, we solve the g2 problem using the eltist RGA with 1 population size and 2 generations. There are 2 variables and the optimal function value is f = RGA improves its best solution slowly towards the end of the run and after 2 generations finds its best solution having f = , indicating the RGA is unable to find the true optimal solution for this problem. Figure 1 shows the variation of exact and approximate KKTPM values of the population-best solution with generations. To show that the true optimal solution 17

18 Figure 9: Generation-wise KKT proximity measure for the population-best RGA solution for Problem g1. Figure 1: Generation-wise KKT proximity measure for the population-best RGA solution for Problem g2. has a zero KKTPM value, we replace 2-th generation RGA solution with the true optimal solution. The figure shows that all the KKTPM values drops to zero at 2-th generation. In this problem, the projected KKTPM approximation is closer to the exact KKTPM values. However, the correlation coefficient between ɛ and ɛest is found to be.976, meaning a strong positive correlation in their changing pattern with generations. Next we consider g4 problem, which has five variables and six constraints besides 1 variable bounds. Figure 11 shows that the elitist RGA with 2, population size run for 5 generations is able to quicly reduce the population-best function value close to nown optimum ( f = ) but get stuc to a near-optimal solution having f = at the 71-st generation. The exact KKTPM value for this solution is.271. However, when this optimal solution is computed for its KKTPM value, it is found to be exactly zero. The estimated KKTPM (ɛ est ) value at the final RGA solution is.18. All approximated KKTPM values are found to be close to the exact KKTPM values. It is also interesting to note that the optimal KKTPM value is always bounded by ɛ D and ɛadj values. The correlation coefficient between ɛ and ɛadj is found to be The problem g6 contains two variables and six constraints, including four variable bounds which are treated as constraints. The elitist RGA is applied with 1, population members and is run for 2, generations. The reason for these large numbers is that for this problem we wanted to achieve a large accuracy. The optimal solution for this problem has a function value of f = Figure 13 shows the variation of different KKTPM values with generation. The elitist RGA finds a solution with f = in 674 generations (having an exact KKTPM value of ) and a solution with f = at 1,898-th generation having an exact KKTPM value of.59. The corresponding estimated KKTPM values at these two solutions are and.73, respectively. The correlation coefficient between ɛ and ɛadj is found to be Problem g7 is a 1-variable, 28-constraint problem. The elitist RGA is run with 1, population members and for 1, generations. The optimal solution of this problem has a function value f = Figure 13 shows the variation of different KKTPM values with generation. Elitist RGA converges to a near-optimal solution having f = This solution 18

19 Figure 11: Generation-wise KKT proximity measure for the population-best RGA solution for Problem g4. Figure 12: Generation-wise KKT proximity measure for the population-best RGA solution for Problem g6. has an exact KKTPM value of.661. The corresponding estimated KKTPM value is.857. The correlation coefficient between ɛ and ɛadj is found to be The problem g8 has two variables and 1 constrains including variable bounds. The elitist RGA is applied to this problem using 2 population members for 1 generations. Figure 14 shows the variation of different KKTPM values with generation. The elitist RGA finds the exact optimum (having f = at 48 generations. All KKTPM values at this generation are zero. The correlation coefficient between ɛ and ɛadj values from all generations of a RGA run is found to be.9985, as also evident from the closeness of these two KKTPM values in the figure Figure 13: Generation-wise KKT proximity measure for the population-best RGA solution for Problem g7. Figure 14: Generation-wise KKT proximity measure for the population-best RGA solution for Problem g8. The next problem g9 is a seven-variable, 18-constraint problem. The elitist RGA is applied using 1, population members and is run for 5 generations. The optimal function value is f = As seen from Figure 15, the elitist RGA converges gradually near this optimal 19

20 solution with generation, but cannot find the true optimal solution until 499 generations. It gets to a solution having f = , which has an exact KKTPM value of.9192, whereas the optimal solution has a KKTPM value of zero. Figure 15 adds the nown optimal solution at 5- th generation to show this fact. Interestingly, for this problem, all approximated KKTPM values are almost identical to each other and to the exact KKTPM value. This can also be explained from the fact that the correlation coefficient of ɛ and ɛadj is found to be one Figure 15: Generation-wise KKT proximity measure for the population-best RGA solution for Problem g9. Figure 16: Generation-wise KKT proximity measure for the population-best RGA solution for Problem g1. The next problem g1 is a challenging eight-variable, 22-constraint problem. The optimal function value for this problem is f = There is a scaling issue of constraint functions for this problem, which maes it difficult to solve this problem to optimality. As shown in Figure 16, the elitist RGA with 2 population members fails to converge near the optimum solution. The KKTPM values are also away from zero even after 5 generations. The best RGA solution has a function value f = (11.3% worse from the optimal function value). The exact KKTPM value at this point is , as shown in the figure, whereas when the optimal solution is deliberately added at the 5-th generation for KKTPM computation, it is calculated to be zero. The corresponding estimated KKTPM value at the best RGA solution is , which is very close to exact KKTPM value. For this problem as well, all approximated KKTPM values are very close to each other. It is not surprising that the correlation coefficient between ɛ and ɛ Adj is found to be one. Problem g18 is a nine-variable, 31-constraint problem. The optimal function value is f = The elitist RGA is run with 1 population members, Figure 17 shows that g18 needs 3 generations to find a feasible solution. At 31-st generation, the best RGA solution has a function value f = with an exact KKTPM value of At generation 199, the population-best near-optimal solution with f = has a KKTPM value of The corresponding estimated KKTPM values are and.1734, respectively. The correlation coefficient between ɛ and ɛadj is found to be The final problem G24 is a two-variable, six constraint problem. The optimal solution has a function value f = The elitist RGA is applied with 2 population size and is run for 1 generations. Figure 18 shows that KKT-proximity measure reduces with generations and reaches to a near-optimal solution quicly. The best RGA solution after 1 generations has a 2

21 Figure 17: Generation-wise KKT proximity measure for the population-best RGA solution for Problem g18. Figure 18: Generation-wise KKT proximity measure for the population-best RGA solution for Problem g24. function value of f = with an exact KKTPM value of The corresponding estimated KKTPM value is Exact and estimated ɛ and ɛadj values follow a similar pattern and the correlation coefficient between them for all generation-wise best solutions is found to be.9925, meaning a strong positive correlation between them Computational Time The extensive results on the above constrained problems indicate that the correlation between the exact KKTPM and the proposed estimated KKTPM is quite strong and close to one. This means that the pattern of variation of these two KKTPM quantities are similar to each other, indicating that the estimated KKTPM can be used to indicate the level of convergence of the population-best solution close to the theoretical optimal solution, without nowing the true optimal solution. As mentioned earlier, the purpose of computing the approximated KKTPM, instead of the exact KKTPM, is the computational advantage which we demonstrated in Table 1. The computational time needed to solve each of the G-series constrained problems with the exact and approximated KKTPM values computed at every generation are presented in seconds of CPU time. It is clear from the table that the optimization based KKTPM computation taes two or three orders of magnitude more time than the approximated methods. The results in above figures and computational time shown in this table together indicates that the proposed estimated KKTPM procedure allows to compute a near-exact KKTPM value within a fraction of computational effort. The final column indicates the computational advantage of the proposed estimated KKTPM compared to the exact KKTPM computational time. As the table shows, the computational advantage varies from about 31 to 781. This is encouraging for the use of the KKTPM for modifying evolutionary or classical optimization methods. While we do not address the possible modifications in this paper, in the next subsection we present results on multi-objective optimization problems Two-Objective ZDT Problems First, we consider commonly used two-objective ZDT problems [29]. ZDT1 has 3 variables and the efficient solutions occur for x i = fori = 2, 3,...,3. NSGA-II is applied with 21