Solving conic optimization problems using MOSEK December 16th 2017
[email protected] www.mosek.com Section 1 Background The company • MOSEK is a Danish company founded 15th of October 1997. • Vision: Create and sell software for mathematical optimization problems. • Linear and conic problems. • Convex quadratically constrained problems. • Also mixed-integer versions of the above. • Located in Copenhagen Denmark at Symbion Science Park. • Daily management: Erling D. Andersen. 2 / 126 Section 2 Outline Todays topics • Conic optimization • What is it? • And why? • What can be modelled? • MOSEK • What is it? • Examples of usage. • Technical details. • Computational results. 4 / 126 Section 3 Conic optimization A special case Linear optimization The classical linear optimization problem: minimize cT x subject to Ax = b; x ≥ 0: Pro: • Structure is explicit and simple. • Data is simple: c; A; b. • Structure implies convexity i.e. data independent. • Powerful duality theory including Farkas lemma. • Smoothness, gradients, Hessians are not an issue. Therefore, we have powerful algorithms and software. 6 / 126 linear optimization cont. Con: • It is linear only. 7 / 126 The classical nonlinear optimization problem The classical nonlinear optimization problem: minimize f(x) subject to g(x) ≤ 0: Pro • It is very general. Con: • Structure is hidden. • How to specify the problem at all in software? • How to compute gradients and Hessians if needed? • How to do presolve? • How to exploit structure e.g. linearity? • Convexity checking! • Verifying convexity is NP hard. • Solution: Disciplined modelling by Grant, Boyd and Ye [6] to assure convexity. 8 / 126 The question Is there a class of nonlinear optimization problems that preserve almost all of the good properties of the linear optimization problem? 9 / 126 Generic conic optimization problem Primal form X minimize (ck)T xk k X subject to Akxk = b; k xk 2 Kk; 8k; where k nk • c 2 R , k m×nk • A 2 R , m • b 2 R , • Kk are convex cones.