A NOTE ON MIXED VARIABLE MATHEMATICAL PROGRAMS

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1 A Note on Mied Variable Mathematical Programs Technical Report RD-5-7 A NOTE ON MIXED VARIABLE MATHEMATICAL PROGRAMS Abstract Isaac Siwale Technical Report No. RD-5-7 Ape Research Limited 65, Southcroft Road London SW6 6QT England ike_siwale@hotmail.com This note presents new benchmark results for some mied integer programming eamples that have recently been reported in the literature. The results were generated by GENO a commercial solver for generalised disjunctive and mied variable non-linear mathematical programs, amongst others types. Key Words: Mied Integer Programming, Generalised Disjunctive Programming, Non-linear Programming. Introduction Recently there has been a marked increase in the development and use of mied integer optimisation models formulated directly or via generalised disjunctive programming techniques, particularly in process systems engineering. Grossmann () reviews the algorithmic techniques that are currently available for solving such models. The purpose of this note is to present new benchmark solutions for some mied integer programming eamples that have recently been reported in the literature. The results were generated by GENO an algorithm that readily solves the more general class of mied variable optimization models. GENO is a real-coded genetic algorithm that can be used to solve uni- or multi-objective optimisation problems. The problems presented may be static or dynamic in character; they may be unconstrained or constrained by equality or inequality constraints, coupled with upper and lower bounds on the variables. The variables themselves may assume real or discrete values in any combination. Although the generic design of the algorithm assumes a multi-objective dynamic optimisation problem, GENO may be specialized for other classes of problems such as the general static optimisation problem, the mied-integer problem, and the two-point boundary value problem, by mere choice of a few parameters. Thus, not only can GENO compute different types of solution to multi-objective problems, it may also be set to generate real or integer-valued solutions, or a miture of the two as required, to uni-objective static and dynamic optimisation problems of varying types. These properties are easily pre-set at the problem set-up stage of the solution process. A detailed description of the algorithm is beyond the scope of this note: rather, the aim here is to demonstrate its capabilities on mied-variable optimisation problems via several numerical eamples as follows. A free trial-version of the program can be obtained by contacting current vendors at: info@aptech.com and sales@tomopt.com Copyright 997 : Ape Research Ltd

2 A Note on Mied Variable Mathematical Programs Technical Report RD-5-7 Eamples The first two eamples present new benchmark solutions for MINLP test problems used by Babu and Angira () and Costa and Oliviera () to test their algorithms; Eample is a mied integer version of a well known test problem; Eample serves to illustrate GENO s capability on the more general mied variable Nonlinear Program; whereas Eample 5 serves to illustrate how generalised disjunctive programs are handled. Eample : [Source: Babu and Angira ()] min J,y (, y) = (y ) + (y ) + (y ) ln(y + ) + ( ) + ( ) + ( ) Subject to: + y + y y y + y. + y.8 + y.5 + y. + y.6 + y.5 + y.6 [, ) y {,} I. GENO Output Generation Objective Optimal Continuous Variable Vector: = (.89, ,.655) T Optimal Discrete Variable Vector: y = (,,, ) T Objective Function Value: J (, y) =.698 This problem was originally proposed by Floudas, et al. (989) and was subsequently tackled by others using various techniques; the latest effort appears to be that by Angira and Babu () who used a differential evolution algorithm. The best known solution has hitherto been as follows: J (, y) =.55766; = (.,.8557,.9556) T ; y = (,,, ) T As can be seen above, the solution by GENO is fundamentally different (note the discrete variable vectors) and significantly better than the best known solution thus far. Copyright 997 : Ape Research Ltd

3 A Note on Mied Variable Mathematical Programs Technical Report RD-5-7 Eample : [Source: Babu and Angira ()] min J ( ) = ( ) Subject to:.9[ ep(.5 )].8[ ep(. )] ( ) ( ) [, ) ; [, ) ; {,} ( ) [ ep(. )].9[ ep(.5 )] I. GENO Output Generation Objective Optimal Variable Vector: Objective Function Value: J () = = (.57,.,.) T This problem was originally proposed by Kocis and Grossmann (989). It is a process synthesis model in which the objective is to select two candidate reactors in order to minimise the production cost. The problem has subsequently been tackled by others using various techniques, and the latest effort appears to be that by Angira and Babu () who used a differential evolution algorithm. The best function value has hitherto been Again, GENO finds a better solution than the best solution known thus far albeit marginally. Copyright 997 : Ape Research Ltd

4 A Note on Mied Variable Mathematical Programs Technical Report RD-5-7 Eample : [Source: Hock and Schittkowski (98, p.)] ( ) = min J 5 Subject to: [78, ] ; [, 5] ; [7, 5] ; [7, 5] ; 5 [7, 5] 5 I. GENO Output Generation Objective Optimal Variable Vector: Objective Function Value: J () = = (78.,., , 5., ) T Both Babu and Angira () and Costa and Oliviera () present an eample similar to this one with some constants slightly different from those above but a significantly different optimal function value. Here however, an MINLP reformulation of the original model is preferred because it has a wider set of comparative solutions. The original source of this problem is reputed to be the Proctor and Gamble Corporation, and the earliest reference appears to be Colville (968). It has featured in many empirical studies on numerical optimisation including Himmelblau (97), Hock and Schittkowski (98), Homaifar, et al. (99), Michalewicz and Fogel (), and Coello Coello (). The best known solution still remains as that reported years ago by Hock and Schittkowski (98) using the Generalised Reduced Gradient (GRG) method. The GRG solution to five decimal places is: = (78.,., 9.995, 5., 6.776) T ; J() = As can be seen from the results above, GENO computes the eact same solution even when treated as a mied integer optimisation problem. Following Costa and Oliviera () the integer variables were taken to be, and Copyright 997 : Ape Research Ltd

5 A Note on Mied Variable Mathematical Programs Technical Report RD-5-7 Eample : [Source: Coello Coello ()] ( ) = min J + Subject to: π π +, 96, [.65, 99] { : =.65N, N Z }, i, ; [., ], i=, = i i i i I. GENO Output Generation Objective Objective Optimal Variable Vector: Objective Function Value: J () = = (.85,.75,.986, ) T This problem has previously been tackled by Deb (997) using GeneAS (Genetic Adaptive Search); by Kannan and Kramer (99) using an augmented Lagrangian multiplier method; and by Sandgren (988) using a branch and bound technique; and by Coello Coello () using a genetic algorithm. Coello Coello (, p.8) presents a comparison of these methods together with his technique: the table below is an etract from there to which has been appended the result by GENO. Coello Coello Deb (997) Kannan, et al. Sandgren GENO Best Function Value As can be seen, the solution by GENO is by far the best amongst those considered; in fact, as of this writing, it is the best known solution. Note also that in the final solution vector, and are integer multiples of.65 as required. Although Hedar and Fukushima (5, p.9) claim to have found a better solution valued , it should be noted that their solution ignores the discreteness restriction on and, and so their algorithm cannot, strictly speaking, be compared to GENO. Legend: Objective is the actual function being minimised; Objective is a merit function for an auiliary program (see Siwale: 6). 5 Copyright 997 : Ape Research Ltd

6 A Note on Mied Variable Mathematical Programs Technical Report RD-5-7 Eample 5: [Source: Lee and Grossmann ()] min J ( ) = 7 Subject to: Y Y Y Where: ( + 5 ) ; ( + ) Y Y Y ( + ) ; ( + 6 ) Y Y ( + 5 ) ; ( ) Y i {,} ; i =, 5, 6 ; Y i {True, False} ; i =,, i [, ) ; i =,,, 7 I. GENO Output Generation Objective Optimal Variable Vector: Objective Function Value: J () =. = (.,.,.,.,.,.,. ) T This is a simple Generalised Disjunctive Program (GDP) whose purpose is to illustrate how GENO may be programmed to solve such problems via the Big-M relaation method. Techniques for converting a GDP into an MINLP are detailed in Raman and Grossmann (99): essentially, logic propositions of the form Y g ( ) i i are replaced by inequalities of the form g ( ) M( y ) ; propositions of the form Y g ( ) are replaced i i by g ( ) My ; and the proposition Y Y Y translates into the inequality y + y + y. The resulting j j MINLP is coded in a straight forward manner with the Big-M parameter simply declared as large a GENO constant which is preset as (see the GAUSS/GENO code listed below). Note that in other solution methods, a judicious choice of the Big-M parameter is imperative because If the value [of M] is too small, then feasible points may be cut off; if [it] is too large, then the continuous relaation might be too loose yielding poor lower bounds (Paraphrased from Lee and Grossmann: 5). Often, it is recommended that Big-M be determined by an auiliary optimisation problem. But with GENO, this step is not necessary: the Big-M parameter needn t be optimised or known in advance. j j The - binary variables used in the Big-M relaation are, 5 and 6. 6 Copyright 997 : Ape Research Ltd

7 A Note on Mied Variable Mathematical Programs Technical Report RD-5-7 // A constrained uni-objective static optimisation problem // Source: Lee and Grossmann () #definecs p_magens 5 #definecs p_popsize #definecs p_agents #definecs p_order 7 #definecs p_plan #include static_gep_defaults.src let vars[p_agents, p_order] = ; let discrete_var[p_order] = ; adj_mode solution_type maimise = "s"; = "e"; = false; timer = true; sol_mt_check = true; constraints_check = true; //cross-over probabilities p_s_over =.55; p_a_over =.55; p_b_over =.; p_h_over =.55; p_d_over =.55; p_shuffle =.; d_factor =.8; quantum_ =.; rand_seed = 657; proc () = m_rate(i,d); retp(.5); endp; proc () = bm_rate(d); retp(.5); endp; //The evaluation function proc () = f(i, d, v_array); local c,fv,u,,z; u = matinit(order, plan, ); = matinit(order, horizon, ); {u,} = assign_sequences(i,d,u,); c = constraints(,,horizon); v_array = evaluate_constraints(c,v_array); fv = objective(,,horizon); retp (fv,v_array); endp; //The objective function proc () = objective(z,,k); local fv; fv = [7,k]; if (maimise); fv = fv; else; fv = -fv; endif; retp(fv); endp; //The functional constraints proc () = constraints(z,,k); local c, M; c = zeros(,); M = large; c[] = [,k] - [,k] M*( - [,k]); c[] = [,k] - [,k] + - M*[,k]; c[] = [,k] - [,k] + - M*( - [5,k]); c[] = [,k] - [,k] M*[5,k]; c[5] = [,k] - [,k] M*( - [6,k]); c[6] = [,k] - [,k] - M*[6,k]; c[7] = [,k] [7,k]; c[8] = [,k] [7,k]; c[9] = [,k] [7,k]; c[] = - [,k] - [5,k] - [6,k]; retp (c); endp; 7 Copyright 997 : Ape Research Ltd

8 A Note on Mied Variable Mathematical Programs Technical Report RD-5-7 Summary This note has presented several numerical eamples solved using GENO a solver for, inter alia, generalised disjunctive and mied variable programs. GENO may easily be programmed to solve generalised disjunctive programs via the Big-M relaation technique. The results for Eamples, and are new bench marks for designers of other algorithms to aim for. Cited References BABU, B. V. and A. R. Angira (). A Differential Evolution Approach for Global Optimisation of MINLP Problems. Proceedings of the Fourth Asia Pacific Conference on Simulated Evolution and Learning (SEAL-),, pp Singapore. COELLO COELLO, C. A. (). Constraint-handling Using an Evolutionary Multi-objective Optimisation Technique. Civil Engineering and Environmental Systems, 7, pp COLVILLE, A. R. (968). A Comparative Study of Non-linear Programming Codes. IBM Scientific Centre Report -99, New York. COSTA L. and P. Oliveira (). Evolutionary Algorithms Approach to the Solution of Mied Integer Non-linear Programming Problems. Computers and Chemical Engineering, 5, pp DEB, K. (997). GeneAS: A Robust Optimal Design Technique for Mechanical Component Design. In D. Dasgupta, and Z. Michalewicz (Eds.). Evolutionary Algorithms in Engineering Applications. Springer-Verlag, Berlin. FLOUDAS, C. A., A. Aggarwal and A. R. Ciric (989). Global Optimum Search for Non-conve NLP and MINLP Problems. Computers and Chemical Engineering,, pp GROSSMANN, I. E. (). Review of Nonlinear Mied Integer and Generalized Conve Disjunctive Programming Techniques. [Online] Available from: [Accessed November, 6]. HEDAR, A. and M. Fukushima. (5). Derivative-Free Filter Simulated Annealing Method for Constrained Continuous Global Optimisation. In G. Di Pillo and F. Giannessi (Eds.). Nonlinear Optimisation and Applications. Kluwer, Amsterdam. HIMMELBLAU, D. (97). Applied Nonlinear Programming, McGraw-Hill, New York. HOCK, W. and K. Schittkowski (98). Test Eamples for Non-linear Programming Codes, Lecture Notes in Economics and Mathematical Systems, 87, Spring-Verlag, Berlin. HOMAIFAR, A. X. Qi and S. H. Lai (99). Constrained Optimisation via Genetic Algorithms. Simulation, 6, pp. -5. KANNAN, B. K. and S. N. Kramer (99). An Augmented Lagrangian Multiplier Based Method for Mied Integer Discrete Continuous Optimisation and its Applications to Mechanical Design. Journal of Mechanical Design. Transactions of the ASME, 6, pp. 8-. KOCIS, G. R. and I. E. Grossmann (989). A Modelling and Decomposition Strategy for the MINLP Optimisation of Process Flow Sheets. Computers and Chemical Engineering,, pp LEE, S. and I. E. Grossmann (). New Algorithms for Generalized Disjunctive Programming. Computers and Chemical Engineering,, pp. 5-. LEE, S. and I. E. Grossmann (5). Logic-based Modelling and Solution of Non-linear Discrete/Continuous Optimisation Problems. Annals of Operations Research, 9, pp MICHALEWICZ, Z. and D. B. Fogel (). How to Solve It. Modern Heuristics. Springer-Verlag, Berlin RAMAN, R. and I. E. Grossmann (99). Relation Between MINLP Modelling and Logical Inference for Chemical Process Synthesis. Computers and Chemical Engineering, 5, pp SANDGREN, E. (988). Nonlinear Integer and Discrete Programming in Mechanical Design. Proceedings of ASME Design Technology Conference, Kissimine, Florida, pp SIWALE, I. (6). GENO TM.: The GAUSS User Manual, th Edition. Technical Report No. RD--5, Ape Research Ltd, London 8 Copyright 997 : Ape Research Ltd