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HANSO and HIFOO Two New MATLAB Codes. Michael L. Overton Courant Institute of Mathematical Sciences New York University Singapore, Jan 2006. Two New Codes. HANSO: Hybrid Algorithm for Nonsmooth Optimization

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Hanso and hifoo two new matlab codes l.jpg


Michael L. Overton

Courant Institute of Mathematical Sciences

New York University

Singapore, Jan 2006

Two new codes l.jpg
Two New Codes

  • HANSO: Hybrid Algorithm for Nonsmooth Optimization

    • Aimed at finding local minimizers of general nonsmooth, nonconvex optimization problems

    • Joint with Jim Burke and Adrian Lewis

  • HIFOO: H-Infinity Fixed-Order Optimization

    • Aimed at solving specific nonsmooth, nonconvex optimization problems arising in control

    • Built on HANSO

    • Joint with Didier Henrion

Many optimization objectives are l.jpg
Many optimization objectives are

  • Nonconvex

  • Nonsmooth

  • Continuous

  • Differentiable almost everywhere

  • With gradients often available at little additional cost beyond that of computing function

  • Subdifferentially regular

  • Sometimes, non-Lipschitz

Numerical optimization of nonsmooth nonconvex functions l.jpg
Numerical Optimization of Nonsmooth, Nonconvex Functions

  • Steepest descent: jams

  • Bundle methods: better, but these are mainly intended for nonsmooth convex functions

  • We developed a simple method for nonsmooth, nonconvex minimization based on Gradient Sampling

  • Intended for continuous functions that are differentiable almost everywhere, and for which the gradient can be easily computed when it is defined

  • User need only write routine to return function value and gradient – and need not worry about nondifferentiable cases, e.g., ties for a max when coding the gradient

  • Very recently, found BFGS is often very effective when implemented correctly - and much less expensive

Slide5 l.jpg

  • Standard quasi-Newton method for optimizing smooth functions, using gradient differences to update an inverse Hessian matrix H, defining a local quadratic model of the objective function

  • Conventional wisdom: jams on nonsmooth functions

  • Amazingly, works very well as long as implemented right (weak Wolfe line search is essential)

  • It builds an excellent, extremely ill-conditioned quadratic model

  • Often, runs until cond(H) is 1016 before breaking down, taking steplengths of around 1

  • Often converges linearly (not superlinearly)

  • By contrast, steepest descent and Newton’s method usually jam

  • Not very reliable, and no convergence theory

Minimizing a product of eigenvalues l.jpg
Minimizing a Product of Eigenvalues

  • An application due to Anstreicher and Lee

  • Min log of product of K largest eigenvalues of AoX

    subject to X positive semidefinite and diag(X)=1

    (A, X symmetric) (we usually set K=N/2)

  • Would be equivalent to SDP if replace product by sum

  • Number of variables is n = N(N-1)/2 where N is dimension of X

  • Following Burer, substitute Y Y’ for X where Y is N by r so n is reduced to rN and normalize Y so its rows have norm 1: thus both constraints eliminated

  • Typically objective is not smooth at minimizer, because the K-th largest eigenvalue is multiple

  • Results for N=10 and rank r = 2 through 10…

N 10 r 6 eigenvalues of optimal a o x l.jpg
N=10, r=6: eigenvalues of optimal AoX











The u and v spaces l.jpg
The U and V Spaces

  • The condition number of H, the BFGS approximation to inverse Hessian, typically reaches 1016 !!

  • Let’s look at eigenvalues of H:

    H = QDQ* (* denotes transpose)

    Search direction is d = -Hg= -QDQ*g (where g = gradient)

  • Next plot shows

    diag(D), sorted into ascending order

    Components of |Q*g|, using same ordering

    Components of |DQ*g|, using same ordering (search direction expanded in eigenvector basis)

  • Tiny eigenvalues of H correspond to “nonsmooth” V-space

  • Other eigenvalues of H correspond to “smooth” U-space

  • From matrix theory, dim(V-space) = m(m+1)/2 – 1, where m is multiplicity of Kth eigenvalue of AoX

More on the u and v spaces l.jpg
More on the U and V Spaces

  • H seems to be an excellent approximation to the “limiting inverse Hessian” and evidently gives us bases for the optimal U and V spaces

  • Let’s look at plots of f along random directions in the U and V spaces, as defined by eigenvectors of H, passing through minimizer (for the N=10, r=6 example)

Line search for bfgs l.jpg
Line Search for BFGS

  • Sufficient decrease:

    for 0 < c1< 1,

    f (x + td) ≤ f (x) + c1 t(grad f)(x)*d

  • Weak Wolfe condition on derivative

    for c1< c2 < 1,

    (grad f)(x + td)*d ≥ c2 (grad f)(x)*d

  • Not strong Wolfe condition on derivative

    | (grad f)(x + t d)*d | ≤ - c2 (grad f)(x)*d

  • Essential when f is nonsmooth

  • No reason to use strong Wolfe even if f is smooth

  • Much simpler to implement

  • Why ever use strong Wolfe?

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But how to check if final x is locally optimal?

  • Extreme ill-conditioning of H is a strong clue, but is expensive to compute and proves nothing

  • If x is locally optimal, running a local bundle method from x establishes nonsmooth stationarity: this involves repeated null-step line searches and building a bundle of gradients obtained via the line searches

  • When line search returns null step, it must also return gradient at a point lying across the discontinuity

  • Then new d = – arg min { ||d||: dÎconvG }, where G is the set of gradients in the bundle

  • Terminate if ||d|| is small

  • If ||d|| is 0 and line search steps were 0, x is Clarke stationary; more realistically it is “close” to Clarke stationary

  • I’ll call this Clarke Quay Stationary since it is a Quay Idea

First order local optimality l.jpg
First-order local optimality

  • Is this implied by Clarke stationarity of f at x ?

  • No, for example f(x) = -|x|is Clarke stationary at 0

  • Yes, in sense that f’(x; d) ≥ 0 for all directions d, when f is regular at x (subdifferentially regular, Clarke regular)

  • Most of the functions that we have studied areregular everywhere, although this is sometimes hard to prove

  • Regularity generalizes smoothness and convexity

Harder problems l.jpg
Harder Problems

  • Eigenvalue product problems are interesting, but the Lipschitz constant L for f is not large

  • Harder problems:

    • Chebyshev Exponential Approximation

    • Optimization of Distance to Instability

    • Pseudospectral Abscissa Optimization (arbitarily large L)

    • Spectral Abscissa Optimization (not even Lipschitz)

  • In all cases, BFGS turns out to be substantially less reliable than gradient sampling

Gradient sampling algorithm initialize e and x repeat l.jpg
Gradient Sampling Algorithm Initialize e and x. Repeat

  • Get G, a set of gradients of function f evaluated at x and at points near x(sampling controlled by e)

  • Let d = – arg min { ||d||: dÎconvG }

  • Line search: replace x by x+ t d, with f(x + t d) < f(x)(if d is not 0)

until d = 0(or ||d|| £ tol)

Then reduce e and repeat.

Convergence theory for gradient sampling method siopt 2005 l.jpg
Convergence Theory for Gradient Sampling Method (SIOPT, 2005)

  • Suppose

    • f is locally Lipschitz and coercive

    • f is continuously differentiable on an open dense subset of its domain

    • number of gradients sampled near each iterate is greater than problem dimension

    • line search uses an appropriate sufficient decrease condition

  • Then, with probability one and for fixed sampling parameter e, algorithm generates a sequence of points with a cluster point xthat is e-Clarke stationary

  • If f has a unique Clarke stationary point x, then the set of all cluster points generated by the algorithm converges to x as e is reduced to zero

  • Kiwiel has already improved on this! (For a slightly different, as yet untested, version of the algorithm.)

Hanso l.jpg
HANSO 2005)

  • User provides routine to compute f and its gradient at any given x (do not worry about nonsmooth cases)

  • BFGS is then run from many randomly generated or user-provided starting points. Quit if gradient at best point found is small.

  • Otherwise, a local bundle method is run from best point found, attempting to verify local stationarity.

  • If this is not successful, gradient sampling is initiated.

  • Regardless, return a final x along with a set of nearby points, the corresponding gradients, and the d which is the smallest vector in the convex hull of these gradients

  • If the nearby points are near enough and d is small enough, f is Clarke Quay Stationary at x

  • This is an approximate first-order local optimality condition if f is regular at x

Optimization in control l.jpg
Optimization in Control 2005)

  • Often, not many variables, as in low-order controller design

  • Reasons:

    • engineers like simple controllers

    • nonsmooth, nonconvex optimization problems in even a few variables can be very difficult

  • Sometimes, more variables are added to the problem in an attempt to make it more tractable

  • Example: adding a Lyapunov matrix variable P and imposing stability on A by AP + PA*£ 0

  • When A depends linearly on the original parameters, this results in a bilinear matrix inequality (BMI), which is typically difficult to solve

  • Our approach: tackle the original problem

  • Looking for locally optimal solutions

H norm of a transfer function l.jpg
H 2005) Norm of a Transfer Function

  • Transfer function G(s)=C (sI – A)-1 B + D

  • SS representation

  • Hnorm is  if A is not stable (stable means all eigenvalues have negative real part) and otherwise, sup||G(s)||2 over all imaginary s

  • Standard way to compute it is norm(SS,inf) in Matlab control toolbox

  • LINORM from SLICOT is much faster

H fixed order controller design l.jpg
H 2005) Fixed-Order Controller Design

  • Choose matrices a, b, c, d to minimize the H norm of the transfer function

  • The dimension of a is k by k (order, between 0 and n=dim(A))

  • The dimension of b is k by p (number of system inputs)

  • The dimension of c is m (number of system outputs) by k

  • The dimension of d is m by p

  • Total number of variables is (k+m)(k+p)

H fixed order controller design32 l.jpg
H 2005) Fixed-Order Controller Design

  • Choose matrices a, b, c, d to minimize the H norm of the transfer function

  • The dimension of a is k by k (order, between 0 and n=dim(A))

  • The dimension of b is k by p (number of system inputs)

  • The dimension of c is m (number of system outputs) by k

  • The dimension of d is m by p

  • Total number of variables is (k+m)(k+p)

  • The case k=0 is static output feedback

H fixed order controller design33 l.jpg
H 2005) Fixed-Order Controller Design

  • Choose matrices a, b, c, d to minimize the H norm of the transfer function

  • The dimension of a is k by k (order, between 0 and n=dim(A))

  • The dimension of b is k by p (number of system inputs)

  • The dimension of c is m (number of system outputs) by k

  • The dimension of d is m by p

  • Total number of variables is (k+m)(k+p)

  • The case k=0 is static output feedback

  • When B1,C1 are empty, all I/O channels are in performance measure

Hifoo h fixed order optimization l.jpg
HIFOO: H 2005) Fixed-Order Optimization

  • Aims to find a, b, c, d for which H norm is locally optimal

  • Begins by minimizing the spectral abscissa max(real(eig(A-block))) until finds a, b, c, d for which A-block is stable (and therefore H norm is finite)

  • Then locally minimizes H norm

  • Calls HANSO to carry out both optimizations

  • Alternative objectives:

    • Optimize the spectral abscissa instead of quitting when stable

    • Optimize the pseudospectral abscissa

    • Optimize the distance to instability (complex stability radius)

  • The output a,b,c,d can then be input to optimize for larger order k, which cannot give worse result

  • Accepts various input formats

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HIFOO Provides the Gradients 2005)

  • For H norm, it combines

    • left and right singular vector info at point on imaginary axis where sup achieved

    • chain rule

  • For spectral abscissa max(real(eig)), it combines

    • left and right eigenvector info for eigenvalue achieving max real part

    • chain rule

  • Do not have to worry about ties

  • Function is continuous, but gradient is not

  • However, function is differentiable almost everyhere (and virtually everywhere that it is evaluated)

  • Typically, not differentiable at an exact minimizer

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Benchmark Examples: AC Suite 2005)

  • Made extensive runs for the AC suite of 18 aircraft control problems in Leibfritz’ COMPLeIB

  • In many cases we found low-order controllers that had smaller H norm than was supposedly possible according to the standard full-order controller MATLAB routine HINFSYN!

  • Sometimes this was even the case when the order was set to 0 (static output feedback)

  • After I announced this at the SIAM Control meeting in July, HINFSYN was extensively debugged and a new version has been released

Benchmark examples mass spring l.jpg
Benchmark Examples: Mass-Spring 2005)

  • A well known simple example with n=4

  • Optimizing spectral abscissa:

    • Order 1, we obtain 0, known to be optimal value

    • Order 2, we obtain about -0.73, previously conjectured to be about -0.5

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Benchmarks: 2005)Belgian Chocolate Challenge

  • Simply stated stabilization problem due to Blondel

  • From 1994 to 2000, not known if solvable by any order controller

  • In 2000, an order 11 controller was discovered

  • We found an order 3 controller, and using our analytical results, proved that it is locally optimal

Availability of hanso and hifoo l.jpg
Availability of HANSO and HIFOO 2005)

  • Version 0.9, documented in a paper submitted to ROCOND 2006, freely available at


  • Version 0.91, available online but not fully tested so no public link yet

  • Version 0.92 will have further enhancements and we hope to announce it widely in a month or two

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Some Relevant Publications 2005)

  • A Robust Gradient Sampling Algorithm for Nonsmooth, Nonconvex Optimization

    • SIOPT, 2005 (with Burke, Lewis)

  • Approximating Subdifferentials by Random Sampling of Gradients

    • MOR, 2002 (with Burke, Lewis)

  • Stabilization via Nonsmooth, Nonconvex Optimization

    • submitted to IEEE Trans-AC (with Burke, Henrion, Lewis)

  • HANSO: A Hybrid Algorithm for Non-Smooth Optimization Based on BFGS, Bundle and Gradient Sampling

    • in planning stage


Gong xi fa cai gung hei fat choy l.jpg

恭喜发财 2005)

Gong xi fa cai!

Gung hei fat choy!