A Global Optimization Approach for the BMI Problem

Proceedings of the 33rd Conference on Decision and Control Lake Buena Vista, FL - December 1994 TP-1 3140 A Global Optimization Approach for the BMI Problem

K.-C. Goh*t M.G. Safonovt 1 G.P. Papavassilopoulos § Dept. of Electrical Engineering - Systems, University of Southern California, Los Angeles, CA 90089-2563. U.S.A.

Abstract Notation will be standard. In particular, for symmetric matrices A and B, X{A} and X{A} refer to the greatest (most The Biaffine Matrix Inequality (BMI) is a potentially very positive) and smallest (most negative) eigenvalues of A, and flexible new framework for approaching complex robust con- A > 0 means A{A} > 0, A > B means A - B > 0. Further trol system synthesis problems with multiple plants, multiple for any vector z E R", llzllco := ma~e{l,,..,,) I.;[. For -a < objectives and controller order constraints. The BMI prob- BL 5 B" < 00, [BL,B"] denotes a closed interval c R. lem may be viewed as the nondifferentiable biconvex pro- This paper will mainly be concerned with obtaining a gramming problem of minimizing the maximum eigenvalue global solution to the following problem: of a biaffine combination of symmetric matrices. The BMI problem is non-local-global in general, i.e. there may exist Definition 1.2 (BMI Eigenvalue Problem) Given the local minima which are not global minima. junction F : X x Y + Rmxmof (I), define While local optimization techniques sometimes yield good results, global optimization procedures need to be considered for the complete solution of the BMI problem. In this paper, we present a global optimization algorithm for the BMI based The BMI Eigenvalue Minimization Problem is on the branch and bound approach. A simple numerical example is included.

Clearly, there exists a solution (z,y) E X x Y to the BMI 1 The Bilinear Matrix Inequality feasibility problem (2) if and only if 0 > min(+,v)EXxyA(I, y). Problem The Linear or Affine Matrix Inequality (LMI/AMI) framework of, e.g. [12, 29, 81, characterized by the problem This paper will be focus on the following problem introduced in [33]: minXEX I{ Fo + ZiFi} (7) Definition 1.1 (The BMI Feasibilty Problem) Given prescribed matrcces F,,3 = FZ RmXm,for z (0,. ,n,}, E E . . where Fi = F:, is a special case of the BMI problem. 3 (0,. . . ,nv}, define the bcafine function F : R"' x R"u E + The properties of the problem (6) were discussed in (181. Rmxm ~ In particular, it is shown that the function h(z,y) is biconuez, n, "9 n= nu i.e. it is convex in z for y fixed and convex in y for I fixed. F(z?y) := &,o + ztF8,0 + yj Fo,3 + zly3F*,3 (1) Further, it is non-local-global in general, i.e. the function ,=I 3=1 *=1 ,=I A(z,y) may have local minima which are not global minima. Fznd, zjzt ezcsts, (I,y) E R"*x RnU such that While to is fairly straightforward to find at least one local minimum of the A(I,y) in X x Y, the complete solution of the F(z,y) < 0 (2) minimization problem, i.e. for some e > 0, find any (Z,$ such For the rest of this paper, restrict (2, y) to some closed bounded that A(S,g) 5 h(z,y) + E for all (z,y) E X x Y, is a global hyper-rectangle X x Y c R"= x R"u where: optimization problem. Global optimization is very hard in general in terms of computational effort required [21,31,25]. x := [B:l,B:l] x ... x [B:nn.,B,u,,] (3) The next section will discuss the background to the BMI problem formulation and hence provide the motivation for Y := [By",,BZ] x . . . x [ByL,u,BK.1 (4) investigating global optimization approaches for completely for some bounds -00 < Bt, 5 Bg < 00, i = 1.. . ,n,, and solving the BMI problem (6). -ca < < B: < 00, j = l...,nv. BLYJ - 'Email: gohOnyquist.usc.edu. 2 Background and Motivation 'Research supported in part by the Air Force Office of Scientific Research, AFOSR grant F49620-92-5-0014, This section will discuss several important robust control syn- *Research supported in part by the Air Force Office of Scientific thesis problems that in general can not be recast as LMI or Research, AFOSR grant F49620-93-1-0505. even convex optimization problems. For these problems the §Research supported in part by the National Science Foundation, NSF grant CCR-9222734. BMI framework offers a viable approach [33, 171. Consider first the p/K,-Synthesis formulation of the rw bust control problem [32,11]. It has been shown [33,17] that

0-7803-1968-0/94$4.0001994 IEEE 2009 the p/K,,,-Synthesis problem for fixed order multipliers and 3 Properties and Local Algorithms a fixed order controller is equivalent to the BMI problem of solving for the matrices T, S, P, W and Q such that the Several important facts about the BME problem are listed. following matrix inequalities hold: See [I81 for details. First note that h(s,y) is nondifferentiable but continuous. However, expressions for its subdifferential at any given (zo,yo) may be obtained. berm{ [ -T w0 ](hT-t UMTQVMT)) > 0 Proposition 3.1 A(s, y) of (6) is Lipschitzion on X x Y. P> 0, - hmm{P(Ro, + Ua,QVa,)} > 0 This follows trivially from the fact that the subdifferential berm{ [ :; 14 > 0 of A(z,y) is uniformly bounded over the bounded domain x x Y. where M,RMT, UMT, VMT, a,, UG,, VQ, are known/prescribed Consider any closed hyper-rectangle Q c X x Y of the matrices. form It is an added bonus that, under mild assumptions, a wide array of robust control synthesis problems may also be expressed in the BMI formulation, e.g., any robust controller synthesis problem consisting of any combinotion of the following may be expressed as a BMI feasibility problem:

0 Multiple objective synthesis, i.e. combinations of p/ Km- Synthesis, 31" and positive real synthesis.

0 Synthesis of one controller for multiple plants, including the Two-Disk 31" problem.

0 Controller synthesis with controller order, structure, and/or stability constraints. Given any Q, define the hyper-rectangle W(Q)C RnrXnV Such problems have been studied previously in, e.g., [5, 23, as follows: 24, 28, 4, 201. Further, the BMI is obviously applicable as a controller parameter tuning framework, particularly within an integrated where wi,j denotes the i,jth element of the matrix W. Define computer-aided control systems analysis and synthesis pack- the osne matrix function of x, y and W, age, e.g. such as the one suggested in [6]. As would be expected, the immenseflexibilty the the BMI fomulation offers has a price in that the resulting optimization problem is no longer convex, quasi-convex or even local- global. However, the lack of such properties does not mean where W E W(Q). Define also that BMI problems are unsolvable. On the contrary, for the BMI: Adz,Y, W):= x{Ft(x,Y, W)) (13) The following result is then obvious: Since the problem is bilinear, one obvious (although suboptimal) way to obtain local solutions is to solve the Proposition 3.2 BMIs as alternating LMIs. There is no reason why a local minimum of A(z, y) with respect to (z,y) jointly cannot be found. The proof follows immediately from the fact that if (.z,y) E Using existing robust controller synthesis and model Q, then the dyad syT E W(Q). reduction techniques, it is possible to generate initial The minimization (6) is very closely related to the LMI conditions which have a high likelihood of being near a eigenvalue minimization problem which has been extensively solution to (2). studied. The approaches based on the interior point methods [26, 22, 7, 35, 151 have bcen particularly succesful. Local minimization procedures coupled with the ability to A straightforward way to use currently available LMI al- make good initial guesses are often sufficenit to give signifi- gorithms to obtain "local solutions" to a BMI problem by al- cantly improved results compared with other currently avail- ternatingly minimizing A(z, y) with respect to z with y fixed able approaches, e.g. see [17]. and vice versa. This approach is not guaranteed to converge It may also be noted the other robust control problems to a stationary point of A(s, y) due to the non-smoothness of or problem formulations which lack the local-global property the function. include the calculation of the multivariable stability margin Another simple extension of LMI techniques to the BMI (Km= t)of a plant, e.g. [lo, 3, 271, LMI problems with rank problem is based on the Method of Centers [22,7]. Given the constraints, e.g. [13, 91, and the optimization over Riccati biaffine matrix function F(s,y) of (I), introduce equation constraints approach of, e.g., [5, 191, for the reduced -log det[al F(z,y)], F(z, > 0 order controller case. - X(a1 - Y)} other wise

2010 Note that the barrier function &(z,y) is convex in (z,a), C1. @L(Q) gives a lower bound and @u(Q)an upper bound and also convex in (y,a). Furthermore, for all (z,y) such on mZn(z,g)~QA(z,Y), i.e., that X{aZ - F(z,y)} > 0, &(z,y) is smooth and at least @dQ) I min A(z,y) I %(e) (16) twice differentiable, with derivatives given in [HI. Algorithm (z.v)EQ 3.1 as defined in Table 1 is then guaranteed to converge to a local minimum of A(z, y). for every hyper-rectangle Q c X x Y. C2. Let Size(Q) denote the length of the longest side of the hyper-rectangle Q, then as Size(Q) \ 0, @u(Q)-. Algorithm 3.1 Method of Centers for BMI [18]. @L(Q) \ 0 uniformly, i.e.,

I. Azc>0,6>OandB~[O,l]. V L > 0 3 6 > 0 such that: V Q c X x Y, Set (do),y(O)) := (O,O), a(') := 6 + A(do),y(O)), k := 0. II. Repeat, { Size(Q) < 6 + @u(Q)- @L(Q) < L (17) Define the following functions on the family of hyper- R1 (dk+l),Y(~+')) := arg minr$,p,(z, y). rectangles of the form Q c X x Y: RI := (1 - B)A(z(k+'),y(k+'))+ oa(k). R3 k:=k+l.

} until A(z(~-'), y("-')) - A(x(~),Y(~)) < L. Clearly, for any Q, @u(Q)may be obtained from Algorithm Table 1: Algorithm 3.1. 3.1. Also, @L(Q)is merely the solution to an LMI problem over Q x W(Q),and can be calculated fairly efficiently [15, In Algorithm 3.1, the minimization in Step R2 is a local 351. minimization with initial point (~(~1,y@)). Note that the local Theorem 4.1 @U and @L as defined in (18) and (19) fulfil minimization in Step R2 guarantees that the solution will conditions C1 and C2. converge to a local minimum of A(r,y). For more complete solution of the BMI problem (6), global Proof: That @U and @L satisfy condition C1 follows optimization techniques need to be examined. Any global op- from Proposition 3.2. timization problem is, of course, inherently difficult. It should From the inequality X{A + B} 5 X{A} +X{B} for hermi- also be noted that in general, global optimization problems tian matrices, and defining vij := v;j(si, yj, wij) = ziyj-wij are NP-hard, e.g. [25]. This, of course, does not mean that the problems are unsolvable, only that they will require alge rithms which will not have polynomial time bounds, unlike, say, LMI or linear programming problems.

4 BMI Branch and Bound Algo- + A(x*, y') - AL(x*, y*, W') 5 I{ 5 5 vtjE,,} rit hm i=l j=1

A branch and bound algorithm for the BMI is presented in where (z*,Y', W') := argmiqz,,)Ee.wEw(e) Adz,Y, W),and this section. Much of the following notation and terminology := vi,j(zf,yjf, w:,~). Note that is from [3, 21 although the global optimization text [21] may also be consulted. Size(Q) 5 6 Before presenting a branch and bound algorithm for the + Size(W(Q)) I S(Bz+ By+6) BMI, it should be note that it is theoretically possible to + vij I 26 (& + By+ a), establish the global optimum to any given tolerence by ez- ViE (1,..., n%},Vj E (1,...,ny) haustiuely gridding the entire domain. This follows from the and define A(Q): fact that A(z, y) is Lipschitz over any bounded domain. The branch and bound approach may be regarded as a clever way A(Q) := @u(Q)-@L(Q) of gridding that uses upper and lower bounds to progressively = min A(z,y) - min h~(z,y,W) refine the areas it needs to grid, therefore avoiding the need (z,u)EQ (z,Y)EQ,WEW(Q) to grid the entire domain. Now, since min(z,,)EQA(z, y) 5 A(z*, y') by definition, and A(Q) 2 0, it follows that 4.1 Upper and Lower Bounds The objective is to minimize A(z, y) over the domain X x Y. The basic requirement for a branch and bound algorithm for minimizing A(z)y) is for the existence of two functions, @L and @U, on the family of hyper-rectangles of the form (8) such that the following conditions hold: so that condition C2 follows. Q.E. D.

201 1 4.2 A BMI Branch and Bound Algorithm Theorem 4.2 Algorithm 4.1 terminates in finite time. Given the functions @u(Q)and QIL(Q), it is straightforward This follows from the fact that @U and @L fulfil conditions to adapt the branch and bound algorithm given in [3, 2) to C1 and C2. An example of the performance of Algorithm globally minimize A(%,y), see Table 2. 4.1 is given in the next section. It should also be noted that if only the BMI feasibility problem is of interest, i.e. if it is desired only to find Algorithm 4.1 Branch and Bound Algorithm for BMI (z,y) ‘such that A(x,y) < 0, then the upper bounds uk may be set to 0 always, and the algorithm is terminated if either I. Fix c > 0. Set k := 0, 80:= X x Y,so := {Qo). %,(Q) < 0 for some Q or if Lk 2 0. CO := @L(QO), U0 := @u(Qo). ZI. Repeat { 4.3 Remarks Rf. Select from Sk such that Lk = @L( 0). The key to Algorithm 4.1 is the development of the lower bound function @L, which exploits the bilinearity of the problem. However, the use of the lower bound of (14) is possibly R2. very conservative. If the geometry of the BMI problem is R9. such that the lower bound of (14) is conservative, then the convergence of Algorithm 4.1 will be slow. Even though convergence to a global minimum is guaranteed in Algorithm 4.1, it may not be practical to apply ;the algorithm to obtain complete solutions to even moder- ate sized robust control synthesis BMI problems due to the R4. large computational load imposed by Algorithm 4.1. How- R5. ever, Algorithm 4.1 is probably more efficient compared to, R6. say, gridding the entire domain of the BMI in question. Note that Algorithm4.1 differs from the branch and bound R 7. approach suggested in (3, 21 in the sense that the upper and lower bounds functions used in [3, 21 are derived from control theory considerations. In contrast, the upper and lower bound functions @U and @L used in Algorithm 4.1 are derived Table 2: Algorithm 4.1 purely from the BMI formulation of (2) itself. The geometrical aspects of the BMI feasibility problem In Algorithm 4.1, sk is the collection of hyper-rectangles were discussed in [HI. In particular, the BMI problem was Qi) after iterations, where At the {Ql,QZ,. . ., k k 5 k. shown to be equivalent to that of finding a hyperplane sep- (IC + 1)th iteration, 0,the rectangle in sk with the smallest arating a given matrix numerical range (field of values) and lower bound, @~(Q)_isidentified, and split along its largest the origin, subject to the constraint that the hyperplane being side into rectangles Q1, and Q2 (Step R2). &+I is formed by generated by a dyad. The connection between that viewpoint discarding e from &. and the lower bound of Proposition 3.2 is obvious. The lower bounds corresponding to ed 01, &, @~(01) A related approach to the BMI problem is that of [34], and @L(&), arz calculated. If @L(Qd > uk, Qi is discarded, which showed that the BMX feasibilty problem may be re- otherwise @u(Q;)is calculated and Q; is added to &+I. duced to the problem of finding the maximum norm element The new estimate for the upper bound, is the low- Uk+1 in an intersection of ellipsoids centered at the origin: a convex est upper bound of the &+I, Step R4. Using Q E see mazimization problem, which again requires global optimiza- is pruned to remove hyper-rectangles for which Uk+l,&+I tion techniques. the global minimum cannot occur, since their lower bounds exceed the current upper bound Uk+l. The pruning step R5 is not strictly necessary, but is required to minimize the number 4.4 Other Global Approaches of hyper-rectangles stored. The new estimate for the lower There are several references in the mathematical program- bound Lk+l is the smallest lower bound of the E Q &+I. ming literature to branch and bound methods for biconvex Note that necessarily, and bilinear problems, e.g., [ 1, 211. Also available is the Ben- ders decomposition (see and references therein) which 0 iP~(01)and @~(&2) cannot be less than Lk, and [16] leads to the primal-relaxed dual algorithm of [14]. The al- 0 Size(Q1) and Size(Q2) cannot exceed Size(@, and the gorithm of [14] is of particular interest. However, it requires “volume” of the rectangles Q1 and Qz will be half that closed form formulae for the gradients of the function to be of Q1. minimized, which is unavailable in the BMI context. i.e. heuristically, it can be Seen that the lower bound Lk and Another obvious approach to the BMI global optimiza- the hyper-rectangle containing the smallest lower bound will tion problem is to use multistart methods, e.g. [30]. Various be successively refined. The details on the finite time con- “intelligent” optimization methods may also be used. How- vergence of the above standard branch and bound algorithm ever, unless the underlying structure of the BMI problem is may be found in, e.g. [Z]. exploited, it is doubtful whether these methods will offer any improvement over Algorithm 4.1.

201 2 Table 3: Local Minima of A(%,y) of (21).

5 A Simple BMI Example Consider a simple BMI problem with its its corresponding LMI: 0 - ...__.__ &.?.?.?.:.: -10 -0.5 -2 -1.8 -0.1 -0.4 -0.5 4.5 0 ] + y [ -0.1 1.2 -1 ] -1 - ...... -2 0 0 -0.4 -1 0 , : __----a

9 0.5 0 002 2 - ...... I ...... ,.: ...... :/.. +I 05 0 -3 +sy 0 -5.5 3 (21) \I ‘j :, [ 0 -3 -11 [2 3 01 3.5 0 0.5 1 1.5 2 -10 -0.5 -2 -1.8 -0.1 -0.4 X -0.5 4.5 0 ] +y [ -0.1 1.2 -1 ] -2 0 0 -0.4 -1 0 Figure 1: Contour Plot of x{F(s,y)}, with trajectories of Al- 9 0.5 0 gorithm 3.1, the local optimization algorithm. Circles denote the local maxima.

For this particular example, the objective isJo minimizeX{F(s, y)} for (s,y)f [-0.5,2] x [-3,7]. Note that F(l,O,l) = -I, so that minX{FL(s,y,w)} = -1 It should first be noted that that there are three local minima, as given in Table 3. Figure 1 gives the contour plots of X{F(s,y)}, and the trajectories of the BMI local optimization algorithm, Alge rithm 3.1. The global optimization algorithm, Algorithm 4.1, required 24 iterations to reduce the difference between the minimum upper bound and the minimum lower bound, uk - Lk to within 0.5% of uk. The eglobal minimum was found to be P = 1.0488 and p = 1.4178, and U24 = A(?, i)= 0.9565. The best lower bound to A(s, y), L2.r = 0.9603. Figure 2 shows the partitions generated by Algorithm 4.1. The endpoints of the local minimization algorithm, Algorithm 3.1, for each partition are also shown. The progress of Algorithm 4.1 is shown in Figure 3. Note that the global minimum is found fairly early, and the re- X mainder of the iterations are devoted to tightening the lower bounds. Figure 2: Contour Plot of x{F(s,y)}, with partitions generated by Algorithm 4.1 shown. Circles are local minimization 6 Summary and Conclusion endpoints. t 0- The Biaffine Matrix Inequality (BMI) formulation is a very flexible framework for approaching complex control system .. synthesis problems. The robust control synthesis problems that fall within the BMI framework include, e.g., the pIK,,,- d Synthesis problem with fixed order multipliers and fixed order ...... controllers, multiobjective synthesis and the synthesis of one .. .. controller for multiple plants. .. The BMI formulation leads to biconvex optimization prob- 30- D s 40 ,s Lo 0 s 70 ,* lems which are often difficult to completely solve, i.e. a local Ilrrs-” ,-*a minimum may not be a global minimum, and it is gener- ally hard to verify that a local minimum is indeed a global Figure 3: Progress of the upper and lower bounds, uk and minimum. While local optimization approaches often yield L,k, and the logarithmic plot of the difference, uk - Lk.

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