Convex Quadratic Programming in AMPL

65 Pages·2013·0.34 MB·English

Convex Quadratic Programming in AMPL
Robert Fourer
Northwestern University / AMPL Optimization
[email protected]
Jared Erickson
Northwestern University / Ziena Optimization
[email protected]
4th International Conference on Continuous Optimization
Caparica, Lisbon, Portugal — 29 July–1 August 2013
Thu.A.23: Extending the Power and Expressiveness of Optimization Modeling Languages
Robert Fourer, Jared Erickson, Convex Quadratic Programming in AMPL 1
ICCOPT 2013 —Lisbon 29 July-1 August 2013
Convex Quadratic Programming in AMPL
A surprising variety of optimization can be identified symbolically and
applications can be written in terms transformed to quadratic problems
of convex quadratic objectives and that have conic formulations. The
constraints that are handled power and convenience of an
effectively by extensions to linear algebraic modeling language may be
solvers. “Elliptical” convex quadratic extended to support these cases, with
programs are easily recognized once the help of a recursive tree-walk
the matrices of quadratic coefficients approach that detects and (where
are extracted, through a test for necessary) transforms arbitrarily
positive-semidefiniteness. “Conic” complex instances; modelers are
problems are also convex quadratic thereby freed from the time-
and can in principle also be detected consuming and error-prone work of
numerically, but are more commonly maintaining the equivalent canonical
recognized by their equivalence to formulations explicitly. We describe
certain canonical forms. Additionally, the challenges of creating the requisite
varied combinations of sums-of- detection and transformation routines
squares, Euclidean norms, quadratic- for the AMPL language, and report
linear ratios, products of powers, computational tests that suggest the
p-norms, and log-Chebychev terms usefulness of these routines.
Robert Fourer, Jared Erickson, Convex Quadratic Programming in AMPL 2
ICCOPT 2013 —Lisbon 29 July-1 August 2013
Outline
What is convex quadratic?
 What kind of solver do you want to use?
Introductory examples
 Product of linear terms
 Traffic network
Detection and transformation
 Where they are done now
 Where they should be done
 Our new extensions
 Theory
 Implementation
 Testing
Robert Fourer, Jared Erickson, Convex Quadratic Programming in AMPL 3
ICCOPT 2013 —Lisbon 29 July-1 August 2013
What is Convex Quadratic?
Convex quadratic objective
 PSD quadratic + linear
Convex quadratic constraints
 Linear
 PSD quadratic ≤ constant
 Conic quadratic
Anything transformable to the above
What kind of solver do you want to use?
Robert Fourer, Jared Erickson, Convex Quadratic Programming in AMPL 4
ICCOPT 2013 —Lisbon 29 July-1 August 2013
What kind of solver?
General Nonlinear Solver
MINOS, KNITRO, Ipopt, SNOPT, CONOPT, . . .
Advantages
 Accepts any form of problem
 Tolerates nonconvexities
Disadvantages
 Relies on smoothness
 Uses complex mechanisms
 Function evals, line searches, convergence tests, . . .
 Reports only local optimality
Robert Fourer, Jared Erickson, Convex Quadratic Programming in AMPL 5
ICCOPT 2013 —Lisbon 29 July-1 August 2013
What kind of solver?
Extended Linear Solver
CPLEX, Gurobi, Xpress, MOSEK, . . .
Advantages
 Uses mechanisms adapted from linear programming
 Sparse coefficient lists, fast interior-point methods, . . .
 Tolerates nonsmooth functions & regions
 Reports global optimality
Disadvantages
 Requires recognizable convex quadratic formulations
 Rejects problems not in required form
Robert Fourer, Jared Erickson, Convex Quadratic Programming in AMPL 6
ICCOPT 2013 —Lisbon 29 July-1 August 2013
⋁
Convex Quadratic Programming in AMPL
Using “Linear” Solvers
Robert Fourer
Northwestern University / AMPL Optimization
[email protected]
Jared Erickson
Northwestern University / Ziena Optimization
[email protected]
4th International Conference on Continuous Optimization
Caparica, Lisbon, Portugal — 29 July–1 August 2013
Thu.A.23: Extending the Power and Expressiveness of Optimization Modeling Languages
Robert Fourer, Jared Erickson, Convex Quadratic Programming in AMPL 7
ICCOPT 2013 —Lisbon 29 July-1 August 2013
Possibilities for Integer Variables
Zero-one
 Extend linear branch-and-bound
 Transform to linear
 requires just one binary in each quadratic term
 many alternatives available
 Transform to PSD quadratic
 based on (cid:1876)(cid:2870) = (cid:1876) for any binary (cid:1876)
General integer
 Extend linear branch-and-bound
 Transform to zero-one
 creates log (cid:1847) binaries for domain of size (cid:1847)
(cid:2870)
Robert Fourer, Jared Erickson, Convex Quadratic Programming in AMPL 8
ICCOPT 2013 —Lisbon 29 July-1 August 2013
Continuous
⋁
Convex Quadratic Programming in AMPL
⋀
Using “Linear” Solvers
Robert Fourer
Northwestern University / AMPL Optimization
[email protected]
Jared Erickson
Northwestern University / Ziena Optimization
[email protected]
4th International Conference on Continuous Optimization
Caparica, Lisbon, Portugal — 29 July–1 August 2013
Thu.A.23: Extending the Power and Expressiveness of Optimization Modeling Languages
Robert Fourer, Jared Erickson, Convex Quadratic Programming in AMPL 9
ICCOPT 2013 —Lisbon 29 July-1 August 2013
Example 1: Product of Linear Terms
Original formulation
(cid:3041) (cid:3041)
 Maximize (∑ (cid:1855) (cid:1876) ) (∑ (cid:1856) (cid:1877) )
(cid:3037)(cid:2880)(cid:2869) (cid:3037) (cid:3037) (cid:3037)(cid:2880)(cid:2869) (cid:3037) (cid:3037)
(cid:3041) (cid:3041)
 ∑ (cid:1855) (cid:1876) ≥ 0, ∑ (cid:1856) (cid:1877) ≥ 0
(cid:3037)(cid:2880)(cid:2869) (cid:3037) (cid:3037) (cid:3037)(cid:2880)(cid:2869) (cid:3037) (cid:3037)
Conic reformulation
 Maximize (cid:1878)
(cid:2870)
 (cid:1878) ≤ (cid:1878) (cid:1878) , (cid:1878) ≥ 0, (cid:1878) ≥ 0
(cid:3051) (cid:3052) (cid:3051) (cid:3052)
(cid:3041) (cid:3041)
 (cid:1878) = ∑ (cid:1855) (cid:1876) , (cid:1878) = ∑ (cid:1856) (cid:1877)
(cid:3051) (cid:3037)(cid:2880)(cid:2869) (cid:3037) (cid:3037) (cid:3052) (cid:3037)(cid:2880)(cid:2869) (cid:3037) (cid:3037)
Robert Fourer, Jared Erickson, Convex Quadratic Programming in AMPL 10
ICCOPT 2013 —Lisbon 29 July-1 August 2013