Global optimization seeks to find the best solution to a

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1 » PRODUCT REVIEW Global Optimization Toolbox for Maple DIDIER HENRION Global optimization seeks to find the best solution to a constrained nonlinear optimization problem by performing a complete search over a set of feasible solutions. In contrast with local optimization, a complete search exhaustively checks the entire feasible region under a Lipschitz-continuity assumption with a known bound on the Lipschitz constant. Some online information on global optimization is available at [1]. As surveyed in [2], many mathematical and engineering problems require a complete search. An example is the 300- year-old Kepler problem of finding the densest packing of equal spheres in three-dimensional Euclidean space, for which a computer-assisted proof is discussed in [3]. The proof involves reducing the problem to several thousand linear programs and using interval calculations to ensure rigorous handling of rounding errors for establishing the correctness of inequalities. Many other difficult problems, such as the traveling salesman and protein folding problems, are global optimization problems. ALGORITHMS The Global Optimization Toolbox (GOT), first released in June 2004 with Maple 9.5, is part of the Maple Professional Toolbox series of add-on products that can be purchased separately. GOT is jointly developed by Maplesoft and Pintér Consulting Services (PCS). PCS is led by János D. Pintér, who has been developing optimization software for over 15 years in collaboration with academic and industrial partners. The core of GOT is the Lipschitz Global Optimizer (LGO), an integrated suite of global and local solvers based on the research summarized in the monograph [4], which was awarded the 2000 INFORMS Computing Society Prize for Research Excellence. A detailed manual is expected to be available by the time this review is published [5]. LGO is at the core of many commercial global optimization packages [6]. Solver engines using LGO are available for C and Fortran compiler platforms, Excel, AIMMS, GAMS, MATLAB (through Tomlab), MPL, and Mathematica. Within GOT, LGO first performs a global search using either a branch and probability bound algorithm, an adaptive stochastic search, or a multistart stochastic search, which is the default option. Constraints are incorporated into a penalty term with a search algorithm that ensures that constraints are satisfied to high precision. After the global search, the solver attempts to refine the solution with a local search based on a generalized reduced-gradient algorithm (a standard technique for local nonlinear optimization) using the approximate solution from the global search as the initial point. If desired, the user can disable the global search phase so that only local search is performed. Recent versions of Maple include an alternative internal optimization package based on the NAG optimization library, which serves exclusively for local optimization. Global convergence of LGO is guaranteed theoretically under the assumption that the objective function and constraints are continuous when using the stochastic methods or Lipschitz continuous when using the branch and bound method [4]. Although optimization problems not satisfying these conditions can still be handled, the quality of the results may vary. For example, since LGO estimates Lipschitz constants numerically using function values only, with genuine black-box knowledge of the objective function, nondifferentiable functions can be handled. The local search phase uses gradient approximations, but does not use the penalty function. First- or higher order derivative information is not used by LGO. FEATURES Besides the well-written online help available under Maple, the main features of GOT are described in [7] through several small, but nontrivial numerical examples. It is immediately apparent that the core LGO solver is perfectly integrated within the Maple environment. In particular, the user can provide optimization model functions and constraints as high-level Maple algebraic, functional, or matrix expressions. See [7, sect. 5] for two illustrative examples of this interaction, namely, minimization of a Bessel function and spline curve fitting. Many other interesting examples are described in the colorful and didactical online example Maple worksheet [8]. Although the input to GOT can be symbolic, the underlying computation is numeric. Maple provides both hardware floating-point arithmetic and software floating-point arithmetic of arbitrary precision; however, GOT uses only hardware arithmetic. Nevertheless, the user can exploit Maple s ability to evaluate the model functions using higher precision arithmetic. 106 IEEE CONTROL SYSTEMS MAGAZINE» OCTOBER X/06/$ IEEE

2 PERFORMANCE The LGO solver was recently benchmarked by Arnold Neumaier s team within the COCONUT project funded by the European Union [9]. The recent survey [2] reports a dozen high-quality solvers for constrained global optimization, including LGO in its GAMS implementation. With the methodology and criteria chosen in [9], LGO is not among the best codes in terms of performance or reliability, although it is one of the few available codes that require only function values. According to Pintér, much better performance in these tests could have been achieved if the ratio between the global and local search efforts was chosen in proportion to the model size, as recommended in the LGO user manual. This automatic parameter tuning is available in both GAMS and GOT but not for the implementation in [9]. See also [10] for a performance evaluation of LGO on a test model set available in GAMS. It is claimed in [7] that, in its current implementations, LGO can handle problems with up to several thousand variables and constraints with corresponding running times on a standard desktop computer ranging from minutes to hours. Recall that the theoretical computational complexity of global optimization algorithms is non-polynomial. Moreover, at fixed dimension, some problem instances can be much more difficult numerically than others. It must also be emphasized that LGO finds only one global optimizer even when multiple optimizers are present. Bound constraints on optimization variables must be explicitly specified when using GOT. As recalled in [9], however, some global optimization packages do not require explicit bounds but rather use bound estimates, which are automatically calculated by the package based on the problem data. Thus, this distinction can be viewed as a limitation of GOT. On the other hand, citing [2, sect. 23], unbounded variables are perhaps the dominant reason for failure of current complete global optimization codes on problems with few variables. One can then argue whether it is reasonable or not to assume that a designer has an a priori good knowledge of bounds on all of the decision variables in a given optimization problem. EXAMPLES We now illustrate the main features of GOT on several nonconvex optimization problems arising in control. Robust Stability Example 1 Our first example is a classical robust stability margin computation problem for a linear system affected by interval uncertainty [11]. This example appears as Example in the optimization benchmark database compiled in [12] and is available for download at titan.princeton. edu/testproblems. The optimization problem reads as min Many difficult problems are global optimization problems. subject to k 10q 2 2 q q3 2 q q2 2 q q3 2 q q 2 q q 1q 2 q q 1q 2 2 q 3 + 1, 000q 2 q q 1 q , 000q2 2 q 3 + 8q 1 q q 1q 2 q 3 q q 1q q 1 q 2 200q 1 0, k q k, 4 2k q k, 6 3k q k, where the decision variables are the scalar parameters k, q 1, q 2, q 3. This problem can be solved with GOT by entering >with(globaloptimization): >GlobalSolve(k, {10*q2^2*q3^3 + 10*q2^3*q3^ *q2^2*q3^ *q2^3*q *q2*q3^3 + q1*q2*q3^2 + q1*q2^2*q *q2*q3^2 + 8*q1*q3^ *q2^2*q3 + 8*q1*q2^2 + 6*q1*q2*q3 - q1^2 + 60*q1*q3 + 60*q1*q2-200*q1 <= 0, -800*k - q1 <= -800, -800*k + q1 <= 800, -2*k - q2 <= -4, -2*k + q2 <= 4, -3*k - q3 <= -6, -3*k + q3 <= 6}, k = , q1 = , q2 = , q3 = ); The first input argument is the objective function to be minimized, the second input argument (the list placed between curly braces) collects the problem constraints, and the remaining input arguments are explicit bounds on the decision variables. Note that it is not possible to omit these bounds. Here we simply choose large intervals. By invoking GOT with these commands, we use the default multistart algorithm. Rounded here to five significant digits, GOT immediately returns the solution k = , q 1 = , q 2 = , and q 3 = , which is the global optimizer reported in [12]. For comparison, we use the Minimize command of the Maple Optimization Package using the same input arguments as above, which yields k = , q 1 = , q 2 = , and q 3 = In this case, a sequential quadratic programming algorithm is used, and a suboptimal solution is returned with k larger by a factor of about 3. This example illustrates the advantage of using GOT over a local optimization algorithm. OCTOBER 2006 «IEEE CONTROL SYSTEMS MAGAZINE 107

3 The Global Optimization Toolbox (GOT), first released in June 2004 with Maple 9.5, is part of the Maple Professional Toolbox series of add-on products that can be purchased separately. Robust Stability Example 2 Example in [12], see also [13], involves five parameters and has the form min subject to k q q 1 w + 16q 2 w + 16w q 1 q 2 78w = 0, 16q 1 w q 1 8q 2 q 3 24w = 0, k q k, k q k, k q k. For this problem, with bounds on all decision variables and its default multistart algorithm, GOT quickly returns the point k = , q 1 = , q 2 = , q 3 = , w = Although this point satisfies the constraints, it is not possible to know whether the solution is globally optimal over this range. This aspect is not specific to GOT, since few global optimization codes provide numerical certificates of global optimality; see [2,sect. 20]. With the additional argument method=branchandbound, the branch-and-bound algorithm of GOT yields the point k = , q 1 = , q 2 = , q 3 = , w = This point satisfies the constraints but yields a 33% larger objective function than the point returned by the multi-start algorithm. Although this solution might be locally optimal, GOT does not provide such information. Next, by refining the search bounds to for all five variables, the multistart algorithm of GOT returns k = , q 1 = , q 2 = , q 3 = , w = , which yields a 10% lower objective function value than the point obtained with larger search bounds. It turns out that this point is the global optimum reported in [12]. It can thus be seen that the solver performance depends on the tightness of the bounds on the decision variables. Fixed-Order Controller Design Next, we seek a fixed-order linear controller maximizing the closed-loop asymptotic decay rate of a linear system. Mathematically speaking, this problem involves spectral abscissa minimization. Given matrices A R n n, B R n m, and C R p n, the problem consists of finding a matrix K R m p that minimizes the spectral abscissa α(a + BKC) defined as the largest real part of the eigenvalues of matrix A + BKC. Note that the condition α(a + BKC) <0 is equivalent to closed-loop asymptotic stabilization. This optimization problem is difficult for at least two reasons. First, as in the two problems considered above, α(a + BKC) is a nonconvex function of K and thus may have multiple local minimizers and stationary points. Second, α(a + BKC) is continuous but, unlike the two problems considered above, is not necessarily differentiable in K [15]. In particular, α(a + BKC) is typically nondifferentiable at local minimizers. Note also that this problem is unconstrained, that is, we have no a priori knowledge of the magnitudes of the entries of a globally optimal matrix K. Moreover, we do not know whether the objective function is bounded below, that is, whether the global minimum α(a + BK C) is finite and attained. We consider a simple but nontrivial example, namely, asymptotic decay rate minimization for the standard twomass-spring benchmark [14] controlled with a secondorder linear controller. Normalized problem data for the closed-loop system are given by A = , B = , C = We note that, if the controller were chosen to be of the order three or four, then the spectral abscissa can be made arbitrarily negative. To apply GOT, we first construct a Maple procedure to compute the spectral abscissa α(a + BKC) given a matrix K. This procedure, which we call specabs, invokes the Eigenvalues function of the LinearAlgebra package, extracts the real parts, and sorts the eigenvalues. LGO is called from GOT with the main command GlobalSolve. In its simplest form for unconstrained optimization, GlobalSolve accepts two input arguments, 108 IEEE CONTROL SYSTEMS MAGAZINE» OCTOBER 2006

4 Control engineering is not among the many existing and potential application areas of the LGO solver implementations listed on the PCS Web site. namely, an objective function to be minimized and a list of lower and upper bounds for all variables. Since the decision variable K is unconstrained, we choose large bounds to begin. For example, we select as the search interval for all variables. As observed for the previous example, this choice may have a significant impact on the performance of LGO. First we run the built-in local search using method=reducedgradient as a third input argument to the command GlobalSolve. After 66 function evaluations, we obtain a spectral abscissa approximately equal to zero. As far as the underlying control problem is concerned, this solution is of a little interest since the resulting controller gain K is not stabilizing. Surprisingly, we obtain the same result using the branch-andbound global search, using method=branchandbound and 2136 function evaluations. With method=singlestart to implement the adaptive random search with a single starting point, GOT can stabilize the plant but convergence of the search is slow. We thus use a timelimit =60 option to stop the solver after 60 s of computation. The best achieved spectral abscissa is This value can be improved with method=multistart using adaptive random search with multiple starting points. After 60 s, the spectral abscissa is pushed further to with K 1 = Relaxing the maximum computation time to 5 min does not significantly improve the result over the same search region. Optimizing over tighter intervals around the estimate K 1 reduces the spectral abscissa to with K 2 = The corresponding closed-loop input step response is satisfactory, with a slight overshoot of 5%, a rise time (required for the step response to rise from 10% to 90% of its final value) of 3.4 s, and a settling time (required for the step response to decline and stay at 5% of its final value) of 9.4 s. Note, however, that we are still quite far from the best known (but not necessarily the globally optimal) spectral abscissa for this example, which, obtained using pole placement methods, is α = ( 15/5) , achieved with, for example, K = Local optimality of this point is proved in [15] using results from nonsmooth variational analysis. We believe that GOT is not able to decrease the spectral abscissa from down to precisely because, as shown in [15], the objective function is a nonsmooth function at local optimizers. Furthermore, in the vicinity of a local optimizer, first-order derivatives of the spectral abscissa are typically very large. Practically speaking, a tiny perturbation around the locally optimal matrix K results in a significant change on the value of α(a + BKC). In other words, the performance achieved by K is highly nonrobust. This feature is typical of an ill-posed optimization problem in control. Indeed, in a realistic control engineering problem, a decay rate constraint is only one of many closed-loop performance requirements in a typical multiobjective optimization setting, together with time-domain, H 2, or H constraints. In summary, on the one hand, GOT cannot find a globally optimal controller because the spectral abscissa function is sensitive around local optima. On the other hand, such a locally optimal controller is highly nonrobust and thus of limited use in practice. CONCLUSIONS Interestingly, control engineering is not among the many existing and potential application areas of the LGO solver implementations listed on the PCS Web site [6]. In fact, one might wonder why global optimization software is not more popular in the control community. Although control-related publications that use global optimization have appeared in the control and global optimization literature, see, for example, [16], [17], and references therein, no genuine computer-aided control system design tools based on global optimization codes have emerged. This situation is in sharp contrast with numerical linear algebra, which is at the core of many control packages (for example, Lyapunov or Riccati equation solvers for MATLAB or Scilab), or, to a lesser extent, convex semidefinite programming over linear matrix inequalities. OCTOBER 2006 «IEEE CONTROL SYSTEMS MAGAZINE 109

5 In summary, the Global Optimization Toolbox for Maple provides a widely applicable, fully integrated development environment that can support control engineering applications and help realize the significant potential that global optimization has as a valuable tool in control system design. ACKNOWLEDGMENTS This review benefited from many constructive comments by Vikram Kapila, Marcel Mongeau, Arnold Neumaier, and János Pintér. Thanks to David Linder of Maplesoft for the core code implementation of function specabs. GOT calculations for the fixed-order controller design problem were carried out by János Pintér. [16] V. Balakrishnan and S.P. Boyd, Global optimization in control system analysis and design, Contr. Dynam. Syst., vol. 53, [17] D. Henrion and J.B. Lasserre, Solving nonconvex optimization problems How GloptiPoly is applied to problems in robust and nonlinear control. IEEE Control Sys. Mag., vol. 24, no. 3, pp , Didier Henrion (henrion@laas.fr) is with LAAS-CNRS, Toulouse, France and the faculty of electrical engineering, Czech Technical University in Prague, Czech Republic. His research interests include numerical algorithms for polynomial matrices, convex optimization over linear matrix inequalities (LMI), robust multivariable control, and computer-aided control system design (CACSD). REFERENCES [1] A. Neumaier, Global optimization, [Online].Available: neum /glopt.html. [2] A. Neumaier, Complete search in continuous global optimization and constraint satisfaction, in Acta Numerica 2004, A. Iserles, Ed. Cambridge, UK: Cambridge Univ. Press, 2004, pp [3] T.C. Hales, A proof of the Kepler conjecture, Ann. Math., vol. 162, pp , [4] J.D. Pintér, Global Optimization in Action., Dordrecht, The Netherlands: Kluwer Academic, [5] J.D. Pintér, Global Optimization with Maple: An INtrocution with Illustrative Examples. Maplesoft, [6] Pintér Consulting Services, [Online]. Available: http// [7] J.D. Pintér, D. Linder, and P. Chin, Global Optimization Toolbox for Maple. An introduction with illustrative applications, Optimiza.Methods Software, vol. 21, no. 4, pp , [8] Maplesoft Worksheet, Applications of the Global Optimization Toolbox, Waterloo Maple Inc., Nov [Online]. Available: [9] A. Neumaier, O. Shcherbina, W. Huyer, and T. Vinkó, A comparison of complete global optimization solvers, Math. Program. B, vol. 103, pp , [10] J.D. Pintér, GAMS/LGO nonlinear solver suite: key features, usage and numerical performance, GAMS Development Corp., Washington, USA, Nov [Online]. [11] R.R.E. de Gaston and M.G. Safonov, Exact calculation of the multiloop stability margin, IEEE Trans. Automat. Control, vol. 33, no. 2. pp , [12] C.A. Floudas and P.M. Pardalos et al., Handbook of Test Problems for Local and Global Optimization. Dordrecht, The Netherlands: Kluwer Academic, [13] J. Ackermann, D. Kaesbauer, and R. Muench, Robust gamma-stability analysis in a plant parameter space, Automatica, vol. 27, no. 1, pp , [14] B. Wie and D.S. Bernstein, Benchmark problems for robust control design, AIAA J. Guidance, Contr. Dynam., vol. 15, no. 5, pp , [15] D. Henrion and M.L. Overton, Maximizing the closedloop asymptotic decay rate for the two-mass-spring control problem, preprint. Albert Einstein on a Bike (6 February 1933) Life is like riding a bicycle. To keep your balance you must keep moving. Albert Einstein (REPRODUCED COURTESY OF THE ARCHIVES, CALIFORNIA INSTITUTE OF TECHNOLOGY) 110 IEEE CONTROL SYSTEMS MAGAZINE» OCTOBER 2006