Advanced Continuous Optimization

Transcription

1 University Paris-Saclay Master Program in Optimization Advanced Continuous Optimization J. Ch. Gilbert (INRIA Paris-Rocquencourt) September 26, 2017 Lectures: September 18, 2017 November 6, 2017 Examination: November 13, 2017 Keep a close eye to Outline 1 Background Convex analysis Nonsmooth analysis Optimization 2 Optimality conditions First order optimality conditions for (P G ) 3 Linearization methods The Josephy-Newton algorithm for inclusions The SQP algorithm in constrained optimization The semismooth Newton method for equations 4 Semidefinite optimization Problem definition Existence of solution and optimality conditions An interior point algorithm 2 / 110

2 Preliminaries Outline of the course Three parts (60h) 1 Advanced Continuous Optimization I (30h) Theory and algorithms in smooth nonconvex optimization Goal: improve the basic knowledge in optimization with some advanced concepts and algorithms Period: September, 18 November, 6 (examination: November, 13) 2 Advanced Continuous Optimization II (20h) Implementation of algorithms (in Matlab on an easy to implement concrete problem) Goal: improve algorithm understanding Period: November, 20 December, 18 (examination: January,??) 3 Specialized course by Jacek Gondzio (10h) 3 / 110 Preliminaries Some features of the course Smooth nonconvex optimization Dealing with the general optimization problem { infx E f(x) (P G ) c(x) G, where f : E R, c : E F, and G is a closed convex set in F. The latter includes the nonlinear optimization problem { infx R f(x) (P EI ) c E (x) = 0 and c I (x) 0, where f : E R, E and I form a partition of [1:m], c E : E R m E, and c I : E R m I are smooth (possibly non convex) functions, the linear semidefinite optimization problem { infx S n C,X (P SDO ) A(X) = b and X 0, where C S n, A : S n R m is linear, and b R m. 4 / 110

3 Preliminaries Some features of the course Finite dimension: E and F are Euclidean vector spaces The unit closed ball is compact A bounded sequence has a converging subsequence Nothing is said about optimal control problems ( see other courses) Emphasis is put on theory optimality conditions of problem (PG ) and many other topics, and algorithms their analysis (scope, convergence, speed of convergence, strong and weak features,... ). 5 / 110 Preliminaries Outline of the course Advanced Continuous Optimization I Anticipated program (we ll see...) 1 Background in convex analysis and optimization Convex analysis: dual cone, Farkas lemma, asymptotic cone,... Nonsmooth analysis: multifunction, subdifferential,... Optimization: optimality conditions,... 2 Abstract optimality conditions First order optimality conditions for (PG ) Second order optimality conditions for (PEI ) (or (P G )?) 3 Linearization methods (Newton) Josephy-Newton algorithm for nonlinear inclusions SQP algorithm for (PEI ) Semismooth Newton algorithm for nonlinear equations 6 / 110

4 Preliminaries Outline of the course Advanced Continuous Optimization I 4 Proximal and augmented Lagrangian methods Proximal algorithm for solving inclusions and optimization problems Augmented Lagrangian algorithm in optimization Application to convex quadratic optimization (error bound and global linear convergence) 5 Semidefinite optimization theory (optimality conditions, duality,... ), algorithm (interior point methods,... ). 7 / 110 Preliminaries Outline of the course Advanced Continuous Optimization II Goal: mastering the implementation of an algorithm Choose one of the following topics (20h) 1 Implementation of an SQP algorithm, application to the hanging chain problem 2 Implementation of an SDO algorithm, application to the OPF problem (Rank relaxation of its QCQP modelling) 8 / 110

5 Background Convex analysis (notation) R := R {,+ }. R + := {t R : t 0} and R ++ := {t R : t > 0}. B, B: open and closed unit balls centered at the origin; for r 0: B(x,r) = x +rb and B(x,r) = x +r B. E, F, G usually denote Euclidean vector spaces. 10 / 110 Background Convex analysis (relative interior I) The affine hull of P E is the smallest affine space containing P: affp := {A : A is an affine space containing P}. The relative interior of P E is its interior in affp: rip := {x P : r > 0 such that [B(x,r) affp] P}. In finite dimension, there holds C convex and nonempty = { ric, affc = aff(ric). 11 / 110

6 Background Convex analysis (relative interior II) Proposition (relative interior criterion) Let C be a nonempty convex set and x E. Then x ric and y C = [x,y[ ric. x ric x 0 C (or affc), t > 1 : (1 t)x 0 +tx C. Let C be a nonempty convex set. Then ri C is convex, C is convex and affc = affc, ri C = C and ri C = ri C (i.e., the last operation prevails). A point x C is said absorbing if d E, t > 0 such that x +td C. x intc x is absorbing. 12 / 110 Background Convex analysis (dual cone and Farkas lemma) The (positive) dual cone of a set P E is defined by P + := {d E : d,x 0, x P}. The (negative) dual cone of a set P is P := P +. Lemma (Farkas, generalized) Let E and F be two Euclidean spaces, A : E F a linear map, and K a nonempty convex cone of E. Then A(K) = {y F : A y K + } +. A : F E is defined by: (x,y) E F, A y,x = y,ax. One cannot get rid of the closure on A(K) in general. If K is polyhedral, then A(K) is polyhedral, hence closed. For K = E, one recovers R(A) = N(A ). 13 / 110

7 Background Convex analysis (tangent and normal cones) Tangent cone Let C be a convex set of E and x C. The cone of feasible directions for C at x is T f x C := R + (C x). The tangent cone to C at x is the closure of the previous one T x C T C (x) = R + (C x). Normal cone Let C be a convex set of E and x C. The normal cone to C at x is There hold N x C N C (x) = {d E : x x,d 0, x C}. N x C = (T x C) and T x C = (N x C). 14 / 110 Background Convex analysis (convex polyhedron) A convex polyhedron in E is a set of the form P = {x E : Ax b}, where A : E R m is a linear map and b R m. It is a closed set. For x P, define I(x) := {i [1:m] : (Ax b) i = 0}. If T : E F is linear, then T(P) is a convex polyhedron. If P 1 and P 2 are polyhedra, then P 1 +P 2 is a polyhedron. T x P = T f x P = {d E : (Ad) I(x) 0}. I(x 1 ) I(x 2 ) = T x1 P T x2 P. N x P = cone{a e i : i I(x)}. I(x 1 ) I(x 2 ) = N x1 P N x2 P. 15 / 110

8 Background Convex analysis (asymptotic cone I) Let E be a vector space of finite dimension and C be a nonempty closed convex set of E. The asymptotic cone of C is C := {d E : C +R + d C} = {d E : C +d C}. Properties C is closed. For any x C: C = {d E : x +R + d C} = t>0 = C x t { d E : {x k } C, {t k } such that x k t k d Boundedness by calculation: C is bounded C = {0}. }. 16 / 110 Background Convex analysis (asymptotic cone II) Two calculation rules (there are many more) If K, then K is a closed convex cone K = K. For an arbitrary collection {Ci } i I of closed convex sets C i with nonempty intersection: ( i I C i ) = i I C i. Example Let A : E F and B : E G be linear maps, a F, b G, K G be a closed convex cone, and P = {x E : Ax = a, Bx b +K}. Then P = {d E : Ad = 0, Bd K}. 17 / 110

9 Background Nonsmooth analysis (multifunction I) A multifunction T (or set-valued mapping) between two sets E and F is a function from E to P(F), the set of the subsets of F. Notation: T : E F : x T(x) F. Same concept as a binary relation (i.e., the data of a part of E F). A graph, the domain, the range of T : E F are defined by G(T) := {(x,y) E F : y T(x)}, D(T) := {x E : (x,y) G(T) for some y F} = π E G(T), R(T) := {y F : (x,y) G(T) for some x E} = π F G(T). The image of a part P E by T is T(P) := x P T(x). 19 / 110 Background Nonsmooth analysis (multifunction II) The inverse of a multifunction T : E F (it always exists!) is the multifunction T 1 : F E defined by T 1 (y) := {x E : y T(x)}. Hence y T(x) x T 1 (y). When E, F are topological/metric spaces, a multifunction T : E F is said to be closed at x E if y T(x) when (xk,y k ) G(T) converges to (x,y), closed if G(T) is closed in E F (i.e., T is closed at any x E). upper semicontinuous at x E if ε > 0, δ > 0, x x +δb, there holds T(x ) T(x)+εB. 20 / 110

10 Background Nonsmooth analysis (multifunction III) When E, F are vector spaces, a multifunction T : E F is said to be convex if G(T) is convex in E F. This is equivalent to saying that (x 0,x 1 ) E 2 and t [0, 1]: Note that T((1 t)x 0 +tx 1 ) (1 t)t(x 0 )+tt(x 1 ). T convex and C convex in E = T(C) convex in F. 21 / 110 Background Optimization (a generic problem) One considers the generic optimization problem { min f(x) (P X ) x X. where f : E R (E is a Euclidean vector space), X is a set of E (possibly nonconvex). 23 / 110

11 Background Optimization (tangent cone to a nonconvex set) A direction d E is tangent to X E at x X if {x k } X, {t k } 0 : x k x t k d. The tangent cone to X at x is the set of tangent directions. It is denoted by T x X or T X (x). Properties Let x X. T x X is closed. X is convex = T x X is convex and T x X = R + (X x). 24 / 110 Background Optimization (Peano-Kantorovich NC1) Theorem (Peano-Kantorovich NC1) If x is a local minimizer of (P X ) and f is differentiable at x, then f(x ) (T x X) +. The gradient of f at x is denoted by f(x) E and is defined from the derivative f (x) by d E : f(x ),d = f (x) d. 25 / 110

12 Background Optimization (NSC1 for convex problems) Recall that a convex function f : E R {+ } has directional derivatives f (x;d) R for all x domf and all d E. Proposition (NSC1 for a convex problem) Suppose that X is convex, f is convex on X, and x X. Then x is a global solution to (P X ) if and only if x X : f (x ;x x ) 0. Proof. Straightforward, using the convexity inequality x X : f(x) f(x )+f (x ;x x ). 26 / 110 Background Optimization (problem (P EI )) A generic form of the nonlinear optimization problem: inf x f(x) (P EI ) c E (x) = 0 c I (x) 0, where f : E R, E and I form a partition of [1:m], c E : E R m E, and c I : E R m I are smooth (possibly non convex) functions. The feasible set is denoted by X EI := {x E : c E (x) = 0, c I (x) 0}. We say that an inequality constraint is active at x X EI if c i (x) = 0. The set of indices of active inequality constraints is denoted by I 0 (x) := {i I : c i (x) = 0}. (P EI ) is said to be convex if f is convex, c E is affine, and c I is componentwise convex. 27 / 110

13 Background Optimization (problem (P EI ) NC1 or KKT conditions) Theorem (NC1 for (P EI ), Karush-Kuhn-Tucker (KKT)) If x is a local minimum of (P EI ), if f and c = (c E,c I ) are differentiable at x, and if c is qualified for representing X EI at x in the sense (2) below, then there exists a multiplier λ R m such that Some explanations. x l(x,λ ) = 0, c E (x ) = 0, The Lagrangian of (P EI ) is the function (1a) (1b) 0 (λ ) I c I (x ) 0. (1c) l : (x,λ) E R m l(x,λ) = f(x)+λ T c(x). The complementarity condition (1c) means (λ ) I 0, (λ ) T I c I (x ) = 0, and c I (x ) / 110 Background Optimization (problem (P EI ) constraint qualification) The tangent cone T x X EI is always contained in the linearizing cone T x X EI := {d E : c E (x) d = 0, c I 0 (x)(x) d 0}. It is said that the constraint c is qualified for representing X EI at x if T x X EI = T x X EI. (2) Sufficient conditions of qualification: continuity and/or differentiability and one of the following (CQ-A) ce I 0 (x) is affine near x (Affinity), (CQ-S) ce is affine, c I 0 (x) is componentwise convex, ˆx X EI such that c I 0 (x)(ˆx) < 0 (Slater), (CQ-LI) i E I 0 (x) α i c i (x) = 0 = α = 0 (Linear Independence), (CQ-MF) i E I 0 (x) α i c i (x) = 0 and α I0 (x) 0 = α = 0 (Mangasarian-Fromovitz). 29 / 110

14 Background Optimization (problem (P EI ) more on constraint qualification) Proposition (other forms of (CQ-MF)) Suppose that c E I0 (x) is differentiable at x X EI. Then the following properties are equivalent: i (CQ-MF) holds at x, ii v R m, d E: c E (x) d = v E and c I 0 (x) (x) d v I 0 (x), iii c E (x) is surjective and d E: c E (x) d = 0 and c I 0 (x)(x) d < / 110 Optimality conditions First order optimality conditions for (P G ) (the problem) [WP.fr] We consider the problem (P G ) { min f(x) c(x) G, where f : E R (E is a Euclidean vector space), c : E F (F is another Euclidean vector space), G is nonempty closed convex set in F, The feasible set is denoted by X G := {x E : c(x) G} = c 1 (G). (P G ) is said to be convex if f is convex, T : E F : x c(x) G is convex (then XG is convex). 32 / 110

15 Optimality conditions First order optimality conditions for (P G ) (tangent and linearizing cones, qualification) Proposition (tangent and linearizing cones) If c is differentiable at x X G, then T x X G T x X G := {d E : c (x) d T c(x) G}. T x X G is called the linearizing cone to X at x. The equality T x X G = T x X G is not guaranteed. The constraint function c is said to be qualified for representing X G at x if T x X G = T x X G, c (x) [(T c(x) G) + ] is closed. (3a) (3b) 33 / 110 Optimality conditions First order optimality conditions for (P G ) (NC1) Theorem (NC1 for (P G )) If x is a local minimum of (P G ), if f and c are differentiable at x, and if c is qualified for representing X G at x in the sense (3a)-(3b), then 1 there exists a multiplier λ F such that f(x )+c (x ) λ = 0, λ N c(x )G. (4a) (4b) 2 if, furthermore, G K is a convex cone, then (4b) can be written K λ c(x ) K or c(x ) N λ K. (4c) One recognizes in (4a) the gradient of the Lagrangian wrt x l : E F R : x l(x,λ) = f(x)+ λ,c(x) and in (4c) the complementarity conditions. 34 / 110

16 Optimality conditions First order optimality conditions for (P G ) (SC1 for convex problem) Theorem (SC1 for convex (P G )) If (P G ) is convex, if f and c are differentiable at x X G, and if there is a λ F such that (x,λ ) satisfies (4a)-(4b), then x is a global minimum of (P G ). The goal of the next slides is to highlight and analyze a condition (Robinson s condition) that 1 provides an error bound for the feasible set y +X G (y small), 2 claims the stability of the feasible set X G, 3 ensures that c is qualified for representing X G. 35 / 110 Optimality conditions First order optimality conditions for (P G ) (Robinson s condition) [WP.fr] We say that Robinson s condition holds at x X G if (CQ-R) We will see below that it is useful since it ( ) 0 int c(x)+c (x) E G. (5) provides an error bound for small perturbations y +XG of X G, claims the stability of the feasible set XG with respect to small perturbations, shows that the constraint function c is qualified for representing XG at x, in the sense (3a)-(3b), it generalizes to (P G ) the Mangasarian-Fromovitz constraint qualification (CQ-MF) for (P EI ). 36 / 110

17 Optimality conditions First order optimality conditions for (P G ) (Robinson s error bound I) [20] Theorem ((CQ-R) and metric regularity) If c is continuously differentiable near x 0 X G, then the following properties are equivalent: 1 (CQ-R) holds at x = x 0, 2 there exists a constant µ 0, such that (x,y) near (x 0,0): dist(x,c 1 (y +G)) µ dist(c(x),y +G). (6) Condition 2 is named metric regularity since it is equivalent to that property (see its definition below) for the multifunction T : E F : x c(x) G. 37 / 110 Optimality conditions First order optimality conditions for (P G ) (Robinson s error bound II) [20] An error bound is an estimation of the distance to a set by a quantity easier to compute. Corollary (Robinson s error bound) If c is continuously differentiable near x 0 X G and if (CQ-R) holds at x = x 0, then there exists a constant µ 0, such that x near x 0 : dist(x,x G ) µ dist(c(x),g). (7) dist(x,x G ) is often difficult to evaluate in E, dist(c(x),g) is often easier to evaluate in F (it is the case if c(x) is easy to evaluate and G is simple), useful in theory (e.g., for proving that (CQ-R) implies constraint qualification), in algorithmics (e.g., for proving [speed of] converge) or in practice (for estimating dist(x,x G )). 38 / 110

18 Optimality conditions First order optimality conditions for (P G ) (stability wrt small perturbations) The estimate (6) readily implies the following stability result. Corollary (stability wrt small perturbations) If c is continuously differentiable near some x 0 X G and if (CQ-R) holds at x 0, then, for all small y F: {x E : c(x) y +G}. 39 / 110 Optimality conditions First order optimality conditions for (P G ) (open multifunction theorem I) [25, 22, 5] Let T : E F be a multifunction and B E and B F be the closed balls of E and F. T is open at (x 0,y 0 ) G(T) with ratio ρ > 0 if there exist a neighborhood W of (x 0,y 0 ) and a radius r max > 0 such that for all (x,y) W G(T) and r [0,r max ]: y +ρr B F T(x +r B E ). T is metric regular at (x 0,y 0 ) G(T) with modulus µ > 0 if for all (x,y) near (x 0,y 0 ): dist(x,t 1 (y)) µ dist(y,t(x)). Difference: (x,y) G(T) for the openness, but not for the metric regularity. y slope ρ G(T) T(x) y +ρ(2r) B F y +ρr B F y dist(y,t(x)) r T(x) x x x 0 y 0 slope 1/µ G(T) T 1 (y) dist(x,t 1 (y)) Both estimate the change in T(x) with x (but these are not infinitesimal notions), either from inside G(T) (openness) or outside (metric regularity). 40 / 110

19 Optimality conditions First order optimality conditions for (P G ) (open multifunction theorem II) [25, 22, 5] Extention of the open mapping theorem for linear (continuous) maps to (nonlinear) convex multifunctions. Theorem (open multifunction theorem, finite dimension) If T : E F is convex and (x 0,y 0 ) G(T), then the following properties are equivalent: 1 y 0 intr(t), 2 for all r > 0, y 0 intt(x +r B E ), 3 T is open at (x 0,y 0 ) with rate ρ > 0, 4 T is metric regular at (x 0,y 0 ) with modulus µ > 0. One can take µ = 1/ρ in point 4 if ρ is given by point / 110 Optimality conditions First order optimality conditions for (P G ) (metric regularity diffusion I) [6] ϕ : E R + is subcontinuous if for all sequences {x k } x such that ϕ(x k ) 0, there holds ϕ(x) = 0. ϕ : E R + is r-steep at x E, with r [0,1[, if for all x B(x,ϕ(x)/(1 r)), there exists x B(x,ϕ(x )), such that ϕ(x ) r ϕ(x ). Lemma (Lyusternik [17, 6]) If ϕ : E R + is subcontinuous and r-steep at x domϕ, with r [0,1[, then ϕ has a zero in B(x,ϕ(x)/(1 r)). 42 / 110

20 Optimality conditions First order optimality conditions for (P G ) (metric regularity diffusion II) [6] Theorem (metric regularity diffusion) Suppose that c : E F a continuous function, G a closed convex set of F, T : E F : x c(x) G is µ-metric regular at (x 0,y 0 ) G(T), δ : E F is Lipschitz near x 0 with modulus L < 1/µ, T : E F : x c(x)+δ(x) G. Then T is also metric regular at (x 0,y 0 +δ(x 0 )) G( T) with modulus µ/(1 Lµ): for all (x,y) near (x 0,y 0 +δ(x 0 )), there holds No need of convexity. dist(x, T 1 (y)) µ 1 Lµ dist(y, T(x)). (8) 43 / 110 M.F. Anjos, J.B. Lasserre (2012). Handbook on Semidefinite, Conic and Polynomial Optimization. Springer. [doi]. A. Ben-Tal, A. Nemirovski (2001). Lectures on Modern Convex Optimization Analysis, Algorithms, and Engineering Applications. MPS-SIAM Series on Optimization 2. SIAM. J.F. Bonnans (1994). Local analysis of Newton-type methods for variational inequalities and nonlinear programming. Applied Mathematics and Optimization, 29, [doi]. J.F. Bonnans, J.Ch. Gilbert, C. Lemaréchal, C. Sagastizábal (2006). Numerical Optimization Theoretical and Practical Aspects (second edition). Universitext. Springer Verlag, Berlin. [authors] [editor] [doi]. J.F. Bonnans, A. Shapiro (2000). Perturbation Analysis of Optimization Problems. Springer Verlag, New York. R. Cominetti (1990). Metric regularity, tangent sets, and second-order optimality conditions. Applied Mathematics and Optimization, 21(1), [doi]. A.L. Dontchev, R.T. Rockafellar (2009). Implicit Functions and Solution Mappings A View from Variational Analysis. Springer Monographs in Mathematics. Springer. F. Facchinei, J.-S. Pang (2003). Finite-Dimensional Variational Inequalities and Complementarity Problems (two volumes). Springer Series in Operations Research. Springer. J.Ch. Gilbert (2015). Éléments d Optimisation Différentiable Théorie et Algorithmes. Syllabus de cours à l ENSTA, Paris. [internet]. S.-P. Han (1976). Superlinearly convergent variable metric algorithms for general nonlinear programming problems. Mathematical Programming, 11, S.-P. Han (1977). A globally convergent method for nonlinear programming. Journal of Optimization Theory and Applications, 22, A.F. Izmailov, M.V. Solodov (2014). Newton-Type Methods for Optimization and Variational Problems. 110 / 110

21 Springer Series in Operations Research and Financial Engineering. Springer. N.H. Josephy (1979). Newton s method for generalized equations. Technical Summary Report 1965, Mathematics Research Center, University of Wisconsin, Madison, WI, USA. N.H. Josephy (1979). Quasi-Newton s method for generalized equations. Summary Report 1966, Mathematics Research Center, University of Wisconsin, Madison, WI, USA. D. Klatte, B. Kummer (2002). Nonsmooth Equations in Optimization - Regularity, Calculus, Methods and Applications. Kluwer Academic Publishers. [doi]. B. Kummer (1988). Newton s method for nondifferentiable functions. In J. Guddat, B. Bank, H. Hollatz, P. Kall, D. Klatte, B. Kummer, K. Lommatzsch, L. Tammer, M. Vlach, K. Zimmerman (editors), Advances in Mathematical Optimization, pages Akademie-Verlag, Berlin. L. Lyusternik (1934). Conditional extrema of functionals. Math. USSR-Sb., 41, L. Qi (1993). Convergence analysis of some algorithms for solving nonsmooth equations. Mathematics of Operations Research, 18, L. Qi, J. Sun (1993). A nonsmooth version of Newton s method. Mathematical Programming, 58, [doi]. S.M. Robinson (1976). Stability theory for systems of inequalities, part II: differentiable nonlinear systems. SIAM Journal on Numerical Analysis, 13, S.M. Robinson (1982). Generalized equations and their solutions, part II: applications to nonlinear programming. Mathematical Programming Study, 19, R.T. Rockafellar, R. Wets (1998). Variational Analysis. Grundlehren der mathematischen Wissenschaften 317. Springer. M. Ulbrich (2011). Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. MPS-SIAM Series on Optimization 11. SIAM Publications, Philadelphia, PA, USA. [doi]. H. Wolkowicz, R. Saigal, L. Vandenberghe (editors) (2000). 110 / 110 Handbook of Semidefinite Programming Theory, Algorithms, and Applications, volume 27 of International Series in Operations Research & Management Science. Kluwer Academic Publishers. J. Zowe, S. Kurcyusz (1979). Regularity and stability for the mathematical programming problem in Banach spaces. Applied Mathematics and Optimization, 5(1), / 110