An Interior Point Solver for Smooth Convex Optimization with An
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An interior p oint solver for smo oth convex optimization with an application to environmental-energy-economic mo dels Olivier Ep elly*y, Jacek Gondzio**, Jean-Philipp e Vial* *Logilab, HEC, University of Geneva 102 Bd. Carl-Vogt, 1211-Geneva, Switzerland ySystems Analysis Section, General Energy Research Department, Paul Scherrer Institute, 5232-Villigen, Switzerland **Department of Mathematics and Statistics University of Edinburgh, King's Buildings, May eld Road, Edinburgh EH9 3JZ, Scotland July, 2000 Abstract Solving large-scale optimization economic mo dels such as markal-macro mo dels proves to be dicult or even out of reach for state-of-the-art solvers. We prop ose an optimizer which takes advantage of their p ossible sp ecial structure: a large dynamic linear blo ck on one side, a small nonlinear convex blo ck on the other one. This framework favors the use of interior p oint metho ds which are ecient for large-scale linear programs and which can handle convex programs. nlphopdm is an implementation of an interior p oint metho d built up on the hopdm co de for linear and convex quadratic optimization [18]. The metho d combines ideas of a globally convergent algorithm [3, 31] and the extension of multiple centrality correctors technique [19] to nonlinear convex programming. It is designed for b eing ho oked to mo deling languages suchasgams and ampl. We present in this pap er preliminary results relative to our researchcode nlphopdm and to commercial nonlinear solvers. Our approachachieves a signi cant computational sp eed- up. This is p erformed via the use of a library which computes exact second derivatives. Interior Point Method; Economic Model; Smooth Convex Optimization 1 1 Intro duction markal-macro [24 ] is a regional planning mo del for the energy sector that is frequently used to supp ort energy p olicy assessment in climate change analysis. The mo del is formulated as a mathematical programming problem, involving a large set of linear constraints and a smaller comp onent of smo oth convex constraints. The main users are economists. They are mostly interested in scenario analyses; sometimes they also emb ed the mo del in a larger framework involving several countries to trace the economic impact of climate change. The to ol they need and favor is an algebraic mo deling language that makes it easy to de ne equations and mo dify data, and which is linked to an ecient and robust optimizer. Presently, the existing markal-macro mo dels are all written in gams [7 ], a p opular mo deling language that is well-suited to the formulation of economic mo dels. gams is linked to four general nonlinear solvers, namely conopt and conopt2 [11 ], minos [26 ] and snopt [17 ]. Currently, the row and column sizes of existing markal-macro mo dels range from 2; 000 3; 000 to 10; 000 15; 000, dep ending on the time mesh and the level of technological details. Even though the sizes are very reasonable by mo dern optimization standards, solving the larger markal-macro problems on p ersonal computers with the ab ove software technology is still a challenge. This is particularly true when the mo del needs to b e solved rep eatedly,as it is the case when several indep endent markal-macro mo dels are comp onents of a larger mo del solved by a decomp osition scheme [5, 6, 8]. In this pap er we present an alternative to the combinations gams/conopt or conopt2, gams/minos or gams/snopt. Namely,wehave develop ed a new optimizer for smo oth convex programming problems, and we have linked it to gams, to preserve ease of implementation for end-users. The algorithm requires the computation of exact rst and second derivatives. Unfortunately, gams do es not compute Hessian comp onents, but other mo deling languages, e.g., ampl [15 ], o er this p ossibility. Since the users may not be eager to translate the gams mo dels into ampl,we prop ose a hybrid solution, mixing gams for the main linear part and ampl for the more restricted nonlinear part. From the user p oint of view, the added complication of using ampl to mo del the nonlinear part is very mo dest, all the more that the syntax in ampl and gams is very similar. The eld of interior p oint metho ds for nonlinear programming has b een rapidly growing in the recent past [2 , 4, 9, 16 , 30 ], giving rise to new solvers such as intopt [16 ], loqo [30 ], mosek [2 ], nitro [9 ] and opinel [4 ]. Our co de may be viewed as one such prop osal. nlphopdm is built up on the primal-dual interior p oint co de hopdm for linear and convex quadratic optimization [18 ] http://www.maths.ed.ac.uk/~gondzio/software/hopdm.html and extends the technique of [31 ]. It incorp orates the linesearch of [31 ] and the higher-order predictor-corrector technique, initially implemented in [1 , 19 ] for linear and convex quadratic problems. We name it nlphopdm thereafter, an acronym for Non-Linear Higher Order Primal-Dual Metho d. Our co de is particularly well suited for problems with a large blo ck of linear constraints. We exploit the two-comp onents structure of markal-macro mo d- els by di erentiating p erturbations of the linear and nonlinear comp onents. This approach achieves a signi cant computational sp eed-up. markal-macro mo dels are currently solved by nlphopdm approximately from 3 to 5 times faster than by minos, from 4 to 70 times faster than by snopt, from 20 to 90 times faster than by conopt2 and from 50 to 370 times faster than by conopt. However, the co de applies to more general smo oth convex problems provided that it is supplied with second derivativevalues, an option that ampl o ers. Our interior p oint implementation nlphopdm is brie y presented in section 2. markal- 2 macro and related economic mo dels are describ ed in section 3. The linking of nlphopdm with mo deling languages, namely ampl and gams, is dealt with in section 4. Lastly,numerical results are presented in section 5, followed by our conclusions. 2 NLPHOPDM: an implementation of an interior p oint metho d nlphopdm applies to problems of the following typ e min f x 1a s.t. Ax = b; 1b g x 0; 1c x 0; 1d n m mn n n l where x 2R ; b 2R ; A 2R , f : R ! R and g : R ! R . We assume that f and g , i =1;::: l, are convex twice continuously di erentiable functions. We further assume that i the problem has a solution and that a constraint quali cation holds at the optimum. nlphopdm is a primal-dual infeasible interior p oint metho d for nonlinear convex pro- grams. It aims at solving the rst order optimality conditions: Ax = b; 2a g x 0; 2b T @g T rf x+A w + x y 0; 2c @x T @g T x y = 0; x 0; 2d X rf x+A w + @x Ygx = 0; y 0; 2e m l where w 2 R and y 2 R are Lagrange multipliers asso ciated with linear and nonlinear constraints, resp ectively. X and Y are diagonal matrices with main diagonals x and y , resp ectively. Actually, nlphopdm intro duces slack variables x and z and aims at solving the following system of equations: Ax = b; 3a g x+s = 0; 3b T @g T x y z = 0; 3c rf x+A w + @x Xz = 0; x 0;z 0; 3d Ys = 0; y 0;s 0: 3e For this purp ose, nlphopdm tracks the solution of a p erturb ed system, in which 3a, 3d and 3e are replaced by the equations Ax = b + r; 4a 1 Xz = ; x 0;z 0; 4b 2 Ys = ; y 0;s 0: 4c 3 m 1 n 2 l There r 2 R is a p erturbation, and 2 R and 2 R are p ositivevectors. Conditions 4a{4c, together with 3b{3c, form the rst order optimality condition of the p erturb ed barrier problem n l X X 1 2 P min f x ln x ln s 5a ; r i j i j i=1 j =1 s.t. Ax = b + r; 5b g x+s =0; 5c x>0;s>0: nlphopdm drives the parameters r; ; to zero to yield an approximate solution of 3 1 2 j of the desired precision. In classical barrier metho ds [13 ], one has r = 0, and the vectors , j =1; 2have equal comp onents. The basic step computes the Newton direction asso ciated with 3b{3c and 4. At the k k k k k current iterate x ;s in the primal space and y ;z ;w in the dual space, nlphopdm solves the linear system 2 3 2 3 2 3 k k +1 A 0 0 0 0 b + r Ax x 6 7 6 7 @g k k k +1 6 7 s g x I 0 0 0 6 7 6 7 s 6 7 @x 6 7 6 7 6 T 7 T k +1 @g 6 7 6 7 @g k k k T k T ; 6 = y 6 7 y + z rf x A w H 0 I A 6 7 6 7 @x @x 6 7 6 7 6 7 k +1 1 k k k k 4 5 z X z 4 Z 0 0 X 0 5 4 5 k +1 2 k k k k w Y s 0 Y S 0 0 P @g 2 2 where the rst derivatives and second derivatives H = r f x+ y r g x are evaluated i i @x k k k k k k at x = x and y = y .