Extended Mathematical Programming Michael C. Ferris University of Wisconsin, Madison Nonsmooth Mechanics Summer School, June 15, 2010 Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 1 / 42
Complementarity Systems Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 2 / 42
Generalized Generalized Equations Suppose T is a maximal monotone operator 0 ∈ F ( z ) + T ( z ) ( GE ) Define P T = ( I + T ) − 1 If T is polyhedral (graph of T is a finite union of convex polyhedral sets) then P T is piecewise affine (continous, single-valued, non-expansive) (GE) is equivalent to 0 = F ( P T ( x )) + x − P T ( x ) and the “path following” algorithm can be defined similarly to the variational inequality case. Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 3 / 42
Optimal Yacht Rig Design Current mast design trends use a large primary spar that is supported laterally by a set of tension and compression members, generally termed the rig Reduction in either the weight of the rig or the height of the VCG will improve performance Design must work well under a variety of weather conditions Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 4 / 42
Complementarity feature Stays are tension only members (in practice) - Hookes Law When tensile load becomes zero, the stay goes slack (low material stiffness) 0 ≥ s ⊥ s − k δ ≤ 0 stay goes slack (low ◮ s axial load ◮ k member spring constant ◮ δ member extension Either s i = 0 or s i = k δ i s: axial load Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 5 / 42
MPCC: complementarity constraints min f ( x , s ) x , s s.t. g ( x , s ) ≤ 0 , 0 ≤ s ⊥ h ( x , s ) ≥ 0 g , h model “engineering” expertise: finite elements, etc ⊥ models complementarity, disjunctions Complementarity “ ⊥ ” constraints available in AIMMS, AMPL and GAMS Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 6 / 42
MPCC: complementarity constraints min f ( x , s ) x , s s.t. g ( x , s ) ≤ 0 , 0 ≤ s ⊥ h ( x , s ) ≥ 0 g , h model “engineering” expertise: finite elements, etc ⊥ models complementarity, disjunctions Complementarity “ ⊥ ” constraints available in AIMMS, AMPL and GAMS NLPEC: use the convert tool to automatically reformulate as a parameteric sequence of NLP’s Solution by repeated use of standard NLP software ◮ Problems solvable, local solutions, hard ◮ Southern Spars Company (NZ): improved from 5-0 to 5-2 in America’s Cup! Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 6 / 42
How to use it? Download “gams” system: google “download gams distribution” Evaluation license provided for “full versions” of PATH, CONOPT, MINOS, MOSEK, NLPEC, MILES, EMP License files available at: http://www.cs.wisc.edu/ ∼ ferris/windows.txt or http://www.cs.wisc.edu/ ∼ ferris/linux.txt or http://www.cs.wisc.edu/ ∼ ferris/mac.txt Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 7 / 42
Extended Mathematical Programs Optimization models improve understanding of underlying systems and facilitate operational/strategic improvements under resource constraints Problem format is old/traditional min x f ( x ) s.t. g ( x ) ≤ 0 , h ( x ) = 0 Extended Mathematical Programs allow annotations of constraint functions to augment this format. Give three examples of this: disjunctive programming, bilevel programming and multi-agent competitive models Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 8 / 42
EMP(i): constraint logic Sequencing Example to minimize makespan: seq(i,j): start(i) + wait(i,j) ≤ start(j) for each pair ( i � = j ), either i before j or j before i empinfo: disjunction * seq(i,j) else seq(j,i) i.e. write down all seq equations, only enforce one of every pair EMP options facilitate either Big M reformulation, or Convex Hull reformulation (Grossmann et al), or CPLEX indicator reformulation Other logic constructs available Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 9 / 42
LogMip: Generalized disjunctive programming min Z ∑ c f ( x ) Objective Function = + k k s.t. r ( x ) 0 Common Constraints ≤ ⎡ ⎤ Y Disjunction jk ⎢ ⎥ OR operator g ( x ) 0 , k K Constraints ∨ ⎢ ≤ ⎥ ∈ j J jk ∈ ⎢ ⎥ k Fixed Charges c γ ⎣ ⎦ = k jk Ω ( ) ( ) Y true = Logic Propositions x R , c R n 1 ∈ ∈ Continuous Variables k Y { { true, false } } ∈ Boolean Variables jk Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 10 / 42
Transmission switching Opening lines in a transmission network can reduce cost Total Cost: $20/MWh x 100 MWh 200 MW generated +$40/MWh x 100 = $6 000/h +$40/MWh x 100 $6,000/h Capacity limit: 100 MW 100 MW generated $20/MWh Capacity limit: 100 MW $20/MWh 133 MW 67 MW 67 MW 33MW 100 MW 200 MW load 200 MW load 33MW g generated 200 MW load 200 MW l d $40/MWh $40/MWh $40/MWh 67 MW 67 MW (a) Infeasible due to line capacity (b) Feasible dispatch Need to use expensive generator due to power flow characteristics and capacity limit on transmission line Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 11 / 42
The basic model c T g min g , f ,θ generation cost g − d = Af , f = BA T θ s.t. A is node-arc incidence θ L ≤ θ ≤ ¯ ¯ θ U bus angle constraints ¯ g L ≤ g ≤ ¯ generator capacities g U ¯ f L ≤ f ≤ ¯ f U transmission capacities with transmission switching (within a smart grid technology) we modify as: c T g min g , f ,θ s.t. g − d = Af θ L ≤ θ ≤ ¯ ¯ θ U ¯ g L ≤ g ≤ ¯ g U f i = ( BA T θ ) i , ¯ f L , i ≤ f i ≤ ¯ either f U , i if i closed or f i = 0 if i open Use EMP to facilitate the disjunctive constraints (several equivalent formulations, including LPEC) Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 12 / 42
Example: Robust Linear Programming Data in LP not known with certainty: min c T x s.t. a T i x ≤ b i , i = 1 , 2 , . . . , m Suppose the vectors a i are known to be lie in the ellipsoids a i ∈ ε i := { ¯ a i + P i u : � u � 2 ≤ 1 } where P i ∈ R n × n (and could be singular, or even 0). Conservative approach: robust linear program min c T x s.t. a T i x ≤ b i , for all a i ∈ ε i , i = 1 , 2 , . . . , m Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 13 / 42
Robust Linear Programming as SOCP/ENLP The constraints can be rewritten as: � � a T b i ≥ sup i x : a i ∈ ε i � � � � a T u T P T a T � P T = ¯ i x + sup i x : � u � 2 ≤ 1 = ¯ i x + i x � � � 2 Thus the robust linear program can be written as � � min c T x s.t. ¯ a T � P T i x + 2 ≤ b i , i = 1 , 2 , . . . , m i x � � � m � min c T x + a T i x , P T ψ C ( b i − ¯ i x ) i =1 where C represents the second-order cone. Our extension allows automatic reformulation and solution (as SOCP) by Mosek or Conopt. Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 14 / 42
EMP(ii): Extended nonlinear programs min x ∈ X f 0 ( x )+ θ ( f 1 ( x ) , . . . , f m ( x )) Examples of different θ least squares, absolute value, Huber function Solution reformulations are very different Huber function used in robust statistics. Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 15 / 42
More general θ functions In general any piecewise linear penalty function can be used: (different upside/downside costs). General form: { y T u − k ( y ) } θ ( u ) = sup y ∈ Y First order conditions for optimality are an MCP! Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 16 / 42
ENLP (Rockafellar): Primal problem min x ∈ X f 0 ( x )+ θ ( f 1 ( x ) , . . . , f m ( x )) “Classical” problem: min exp( x 1 ) x 1 , x 2 , x 3 s.t. log( x 1 ) = 1 x 2 2 ≤ 2 x 1 / x 2 = log( x 3 ) , 3 x 1 + x 2 ≤ 5 , x 1 ≥ 0 , x 2 ≥ 0 Soft penalization of red constraints: exp( x 1 )+5 � log( x 1 ) − 1 � 2 + 2 max( x 2 min 2 − 2 , 0) x 1 , x 2 , x 3 s.t. x 1 / x 2 = log( x 3 ) , 3 x 1 + x 2 ≤ 5 , x 1 ≥ 0 , x 2 ≥ 0 Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 17 / 42
ENLP: Primal problem min x ∈ X f 0 ( x )+ θ ( f 1 ( x ) , . . . , f m ( x )) x ∈ R 3 : 3 x 1 + x 2 ≤ 5 , x 1 ≥ 0 , x 2 ≥ 0 � � X = f 1 ( x ) = log( x 1 ) − 1 , f 2 ( x ) = x 2 2 − 2 , f 3 ( x ) = x 1 / x 2 − log( x 3 ) θ 1 ( u ) = 5 � u � 2 , θ 2 ( u ) = 2 max( u , 0) , θ 3 ( u ) = ψ { 0 } ( u ) θ nonsmooth due to the max term; θ separable in example. θ is always convex. Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 18 / 42
Specific choices of k and Y { y ′ u − k ( y ) } θ ( u ) = sup y ∈ Y 4 λ y 2 , Y = ( −∞ , + ∞ ) 1 L 2 : k ( y ) = L 1 : k ( y ) = 0, Y = [ − ρ, ρ ] L ∞ : k ( y ) = 0, Y = ∆, unit simplex 1 4 λ y 2 , Y = [ − ρ, ρ ] Huber: k ( y ) = Second order cone constraint: k ( y ) = 0, Y = C ◦ Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 19 / 42
Elegant Duality For these θ (defined by k ( · ) , Y ), duality is derived from the Lagrangian: L ( x , y ) = f 0 ( x ) + � m i =1 y i f i ( x ) − k ( y ) x ∈ X , y ∈ Y Dual variables in Y not simply ≥ 0 or free. Saddle point theory, under convexity. Dual Problem and Complete Theory. Special case: ELQP - dual problem is also an ELQP. Ferris (Univ. Wisconsin) EMP Nonsmooth School, June 2010 20 / 42
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