Lecture 13: Constrained optimization

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Rodger Kelly
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2 Basic ideas A nonlinearly constrained problem must somehow be converted relaxed into a problem which we can solve (a linear/quadratic or unconstrained problem) We solve a sequence of such problems To make sure that we tend towards a solution to the original problem, we must impose properties of the original problem more and more How is this done? In simpler problem like linearly constrained ones, a line search in f is enough For more general problems, where (normally) the constraints are manipulated, this is not enough We can include penalty functions for constraints that we relax We can produce estimates of the Lagrange multipliers and invoke them We will look at both types of approaches

3 Penalty functions, I Consider the optimization problem to minimize f (x), subject to x S, (1) where S R n is non-empty, closed, and f : R n R is differentiable Basic idea behind all penalty methods: to replace the problem (1) with the equivalent unconstrained one: minimize f (x) + χ S (x), where χ S (x) = { 0, if x S, +, otherwise is the indicator function of the set S

4 Penalty functions, II Feasibility is top priority; only when achieving feasibility can we concentrate on minimizing f Computationally bad: non-differentiable, discontinuous, and even not finite (though it is convex provided S is convex). Better: numerical warning before becoming infeasible or near-infeasible Replace the indicator function with a numerically better behaving function

5 Exterior penalty methods, I SUMT (Sequential Unconstrained Minimization Techniques) devised in the late 1960s by Fiacco and McCormick; still among the more popular ones for some classes of problems, although there are later modifications that are more often used Suppose S = {x R n g i (x) 0, h j (x) = 0, i = 1,...,m, j = 1,...,l}, g i C(R n ), i = 1,...,m, h j C(R n ), j = 1,...,l Choose a C 0 function ψ : R R + such that ψ(s) = 0 if and only if s = 0 [typical examples of ψ( ) will be ψ 1 (s) = s, or ψ 2 (s) = s 2 ]. Approximation to χ S : ( m ν ˇχ S (x) := ν ψ ( max{0,g i (x)} ) l + ψ ( h j (x) )) i=1 j=1

6 Exterior penalty methods, II ν > 0 is a penalty parameter Different treatment of inequality/equality constraints since an equality constraint is violated whenever h j (x) 0, while an inequality constraint is violated only when g i (x) > 0; equivalent to max{0,g i (x)} 0 ˇχ S approximates χ S from below (ˇχ S χ S )

7 Example Let S = {x R 2 x 2 0,(x 1 1) 2 + x 2 2 = 1 } Let ψ(s) = s 2. Then, ˇχ S (x) = [max{0, x 2 }] 2 + [(x 1 1) 2 + x 2 2 1] 2 Graph of ˇχ S and S:

8 Properties of the penalty problem Assume (1) has an optimal solution x Assume that for every ν > 0 the problem to has at least one optimal solution x ν minimize f (x) + ν ˇχ S (x) (2) x R n ˇχ S 0; ˇχ S (x) = 0 if and only if x S The Relaxation Theorem states that the inequality f (x ν ) + ν ˇχ(x ν ) f (x ) + χ S (x ) = f (x ) holds for every positive ν (lower bound on the optimal value) The problem (2) is convex if (1) is

9 The algorithm and its convergence properties, I Assume that the problem (1) possesses optimal solutions. Then, as ν + every limit point of the sequence {x ν } of globally optimal solutions to (2) is globally optimal in the problem (1) Of interest only for convex problems. What about general problems?

10 The algorithm and its convergence properties, II Let f, g i (i = 1,...,m), and h j (j = 1,...,l), be in C 1 Assume that the penalty function ψ is in C 1 and that ψ (s) 0 for all s 0 Then: x k stationary in (2) x k ˆx as k + = ˆx stationary (KKT) in (1) LICQ holds at ˆx ˆx feasible in (1) From the proof we obtain estimates of Lagrange multipliers: the optimality conditions of (2) gives that µ i ν k ψ [max{0,g i (x k )}] and λ j ν k ψ [h j (x k )]

11 Interior penalty methods, I In contrast to exterior methods, interior penalty, or barrier, function methods construct approximations inside the set S and set a barrier against leaving it If a globally optimal solution to (1) is on the boundary of the feasible region, the method generates a sequence of interior points that converge to it We assume that the feasible set has the following form: S = {x R n g i (x) 0, i = 1,...,m } We need to assume that there exists a strictly feasible point ˆx R n, i.e., such that g i (ˆx) < 0, i = 1,...,m

12 Interior penalty methods, II Approximation of χ S (from above, that is, ˆχ S χ S ): { ν m i=1 ν ˆχ S (x) := φ[g i(x)], if g i (x) < 0, i = 1,...,m, +, otherwise, where φ : R R + is a continuous, non-negative function such that φ(s k ) for all negative sequences {s k } converging to zero Examples: φ 1 (s) = s 1 ; φ 2 (s) = log[min{1, s}] The differentiable logarithmic barrier function φ 2 (s) = log( s) gives rise to the same convergence theory, if we drop the non-negativity requirement on φ Barrier function convex if (1) is

13 Example Consider S = {x R x 0 }. Choose φ = φ 1 = s 1. Graph of the barrier function ν ˆχ S in below figure for various values of ν (note how ν ˆχ S converges to χ S as ν 0!): ν=1 ν=0.1 ν= χ S x

14 Algorithm and its convergence Penalty problem: minimize f (x) + ν ˆχ S (x) (3) Convergence of global solutions to (3) to globally optimal solutions to (1) straightforward. Result for stationary (KKT) points more practical: Let f and g i (i = 1,...,m), an φ be in C 1, and that φ (s) 0 for all s < 0 Then: x k stationary in (3) x k ˆx as k + = ˆx stationary (KKT) in (1) LICQ holds at ˆx If we use φ(s) = φ 1 (s) = 1/s, then φ (s) = 1/s 2, and the sequence {ν k /gi 2(x k)} converges towards the Lagrange multiplier ˆµ i corresponding to the constraint i (i = 1,...,m)

15 Interior point (polynomial) method for LP, I Consider the dual LP to maximize b T y, subject to A T y + s = c, s 0 n, (4) and the corresponding system of optimality conditions: A T y + s = c, Ax = b, x 0 n, s 0 n, x T s = 0

16 Interior point (polynomial) method for LP, II Apply a barrier method for (4). Subproblem: n minimize b T y ν log(s j ) j=1 subject to A T y + s = c The KKT conditions for this problem is: A T y + s = c, Ax = b, x j s j = ν, j = 1,...,n (5) Perturbation in the complementary conditions!

17 Interior point (polynomial) method for LP, II Using a Newton method for the system (5) yields a very effective LP method. If the system is solved exactly we trace the central path to an optimal solution, but polynomial algorithms are generally implemented such that only one Newton step is taken for each value of ν k before it is reduced A polynomial algorithm finds, in theory at least (disregarding the finite precision of computer arithmetic), an optimal solution within a number of floating-point operations that are polynomial in the data of the problem

18 Sequential quadratic programming (SQP) methods, I We study the equality constrained problem to minimize f (x), (6a) subject to h j (x) = 0, j = 1,...,l, (6b) where f : R n R and h j : R n R are in C 1 on R n The KKT conditions state that at a local minimum x of f over the feasible set, where x satisfies some CQ, there exists a vector λ R l with l x L(x,λ ) := f (x ) + λ j h j(x ) = 0 n, j=1 λ L(x,λ ) := h(x ) = 0 l Appealing to find a KKT point by directly attacking this system of nonlinear equations, which has n + l unknowns as well as equations

19 Sequential quadratic programming (SQP) methods, II Newton s method! Suppose f and h j (j = 1,...,l) are in C 2 on R n and we have an iteration point (x k,λ k ) R n R l Next iterate (x k+1,λ k+1 ) := (x k,λ k ) + (p k,v k ), where (p k,v k ) R n R l solves the second-order approximation of the stationary point condition for the Lagrange function: 2 L(x k,λ k )( pk v k that is, [ 2 xx L(x k,λ k ) h(x k ) h(x k ) T 0 m m ) = L(x k,λ k ), ]( pk v k ) = ( ) x L(x k,λ k ) h(x k ) Interpretation: the KKT system for the QP problem to 1 minimize p 2 pt 2 xxl(x k,λ k )p + x L(x k,λ k )p, (8a) subject to h j (x k ) + h j (x k ) T p = 0, j = 1,...,l (8b) (7)

20 Sequential quadratic programming (SQP) methods, III Objective: second-order approximation of the Lagrange function with respect to x. Constraints: first-order approximations at x k. The vector v k appearing in (7) is the vector of Lagrange multipliers for the constraints (8b) Unsatisfactory: 1 Convergence is only local 2 The algorithm requires strong assumptions about the problem

21 Introducing an exact penalty function Consider (1), and m l ˇχ S (x) := maximum {0,g i (x)} + h j (x), i=1 j=1 P e (x) := f (x) + ν ˇχ S (x) Suppose x is a KKT point for (1), with Lagrange multipliers (µ,λ ), and (1) is a convex problem Then: if the value of ν is large enough such that ν maximum{µ i, i I(x ); λ j, j = 1,...,l} then the vector x is also a global minimum of the function P e

22 Basic SQP method Given x k R n and a vector (µ k,λ k ) R m + Rl, choose a positive definite, symmetric matrix B k R n n Solve minimize p 1 2 pt B k p + f (x k ) T p, (9a) subject to g i (x k ) + g i (x k ) T p 0, i = 1,...,m, h j (x k ) + h j (x k ) T p = 0, j = 1,...,l (9b) (9c) If B k 2 xx L(x k,µ k,λ k ) then (9) is a 2 nd order approximation of KKT (cf. quasi-newton!)

23 Convergence One can show that if the penalty value is large enough, then the vector p k from (9) is a direction of descent with respect to the exact penalty function P e at (x k,µ k,λ k ) Convergence of the SQP method towards KKT points can then be established under additional conditions on the choices of matrices {B k }

24 Remarks Selecting the value of ν is difficult No guarantees that the subproblems (9) are feasible; we assumed above that the problem is well-defined P e is only continuous; some step length rules infeasible Fast convergence not guaranteed (the Maratos effect) Penalty methods in general suffer from ill-conditioning. For some problems the ill-conditioning is avoided Exact penalty SQP methods suffer less from ill-conditioning, and the number of QP:s needed can be small. They can, however, cost a lot computationally fmincon in MATLAB is an SQP-based solver

25 Filter-SQP Popular development: algorithms where the penalty parameter is avoided altogether filter-sqp methods Multi-objective optimization: x 1 dominates x 2 if ˇχ(x 1 ) ˇχ(x 2 ) and f (x 1 ) f (x 2 ) (if x 1 is better in terms of feasibility and optimality) Filter: a list of pairs (ˇχ i,f i ) such that ˇχ i < ˇχ j or f i < f j for all j i in the list Its elements build up an efficient frontier in the bi-criterion problem Filter used in place of the penalty function, when the standard Newton-like step cannot be computed

Algorithms for constrained local optimization

Algorithms for constrained local optimization Fabio Schoen 2008 http://gol.dsi.unifi.it/users/schoen Algorithms for constrained local optimization p. Feasible direction methods Algorithms for constrained