OPTIMIZATION OVER SYMMETRIC CONES UNDER UNCERTAINTY By BAHA' M. ALZALG A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY WASHINGTON STATE UNIVERSITY Department of Mathematics DECEMBER 2011 To the Faculty of Washington State University: The members of the Committee appointed to examine the dissertation of BAHA' M. ALZALG find it satisfactory and recommend that it be accepted. K. A. Ariyawansa, Professor, Chair Robert Mifflin, Professor David S. Watkins, Professor ii For all the people iii ACKNOWLEDGEMENTS My greatest appreciation and my most sincere \Thank You!" go to my advisor Pro- fessor Ari Ariyawansa for his guidance, advice, and help during the preparation of this dissertation. I am also grateful for his offer for me to work as his research assistant and for giving me the opportunity to write papers and visit several conferences and workshops in North America and overseas. I wish also to express my appreciation and gratitude to Professor Robert Mifflin and Professor David S. Watkins for taking the time to serve as committee members and for various ways in which they helped me during all stages of doctoral studies. I want to thank all faculty, staff and graduate students in the Department of Mathe- matics at Washington State University. I would especially like to thank from the faculty Associate Professor Bala Krishnamoorthy, from the staff Kris Johnson, and from the students Pietro Paparella for their kind help. Finally, no words can express my gratitude to my parents and my grandmother for love and prayers. I also owe a special gratitude to my brothers and sisters, and some relatives in Jordan for their support and encouragement. iv OPTIMIZATION OVER SYMMETRIC CONES UNDER UNCERTAINTY Abstract by Baha' M. Alzalg, Ph.D. Washington State University December 2011 Chair: Professor K. A. Ariyawansa We introduce and study two-stage stochastic symmetric programs (SSPs) with recourse to handle uncertainty in data defining (deterministic) symmetric programs in which a linear function is minimized over the intersection of an affine set and a symmetric cone. We present a logarithmic barrier decomposition-based interior point algorithm for solving these problems and prove its polynomial complexity. Our convergence analysis proceeds by showing that the log barrier associated with the recourse function of SSPs behaves as a strongly self-concordant barrier and forms a self-concordant family on the first stage solutions. Since our analysis applies to all symmetric cones, this algorithm extends Zhao's results [48] for two-stage stochastic linear programs, and Mehrotra and Ozevin's¨ results [25] for two-stage stochastic semidefinite programs (SSDPs). We also present another class of polynomial-time decomposition algorithms for SSPs based on the volumetric barrier. While this extends the work of Ariyawansa and Zhu [10] for SSDPs, our analysis is based on utilizing the advantage of the special algebraic structure associated with the symmetric cone not utilized in [10]. As a consequence, we are able to significantly simplify the proofs of central results. We then describe four applications leading to the SSP problem where, in particular, the underlying symmetric cones are second-order cones and rotated quadratic cones. v Contents Acknowledgement iv Abstract v Chapter 1 1 Introduction and Background 1 1.1 Introduction . .1 1.2 What is a symmetric cone? . .7 1.3 Symmetric cones and Euclidean Jordan algebras . 11 2 Stochastic Symmetric Optimization Problems 28 2.1 The stochastic symmetric optimization problem . 28 2.1.1 Definition of an SSP in primal standard form . 29 2.1.2 Definition of an SSP in dual standard form . 30 2.2 Problems that can be cast as SSPs . 31 3 A Class of Polynomial Logarithmic Barrier Decomposition Algorithms for Stochastic Symmetric Programming 42 3.1 The log barrier problem for SSPs . 43 3.1.1 Formulation and assumptions . 43 2 3.1.2 Computation of rxηe(µ, x) and rxxηe(µ, x).............. 50 vi 3.2 Self-concordance properties of the log-barrier recourse . 53 3.2.1 Self-concordance of the recourse function . 53 3.2.2 Parameters of the self-concordant family . 59 3.3 A class of logarithmic barrier algorithms for solving SSPs . 64 3.4 Complexity analysis . 65 3.4.1 Complexity for short-step algorithm . 66 3.4.2 Complexity for long-step algorithm . 67 4 A Class of Polynomial Volumetric Barrier Decomposition Algorithms for Stochastic Symmetric Programming 75 4.1 The volumetric barrier problem for SSPs . 76 4.1.1 Formulation and assumptions . 76 4.1.2 The volumetric barrier problem for SSPs . 77 2 4.1.3 Computation of rxη(µ, x) and rxxη(µ, x).............. 80 4.2 Self-concordance properties of the volumetric barrier recourse . 85 4.2.1 Self-Concordance of η(µ, ·)....................... 86 4.2.2 Parameters of the self-concordant family . 93 4.3 A class of volumetric barrier algorithms for solving SSPs . 97 4.4 Complexity analysis . 98 4.4.1 Complexity for short-step algorithm . 100 4.4.2 Complexity for long-step algorithm . 101 5 Some Applications 106 5.1 Two applications of SSOCPs . 106 5.1.1 Stochastic Euclidean facility location problem . 106 5.1.2 Portfolio optimization with loss risk constraints . 112 5.2 Two applications of SRQCPs . 118 5.2.1 Optimal covering random ellipsoid problem . 118 vii 5.2.2 Structural optimization . 125 6 Related Open problems: Multi-Order Cone Programming Problems 130 6.1 Multi-order cone programming problems . 131 6.2 Duality . 134 6.3 Multi-oder cone programming problems over integers . 138 6.4 Multi-oder cone programming problems under uncertainty . 139 6.5 An application . 140 6.5.1 CERFLPs|An MOCP model . 142 6.5.2 DERFLPs|A 0-1MOCP model . 143 6.5.3 ERFLPs with integrality constraints|An MIMOCP model . 145 6.5.4 Stochastic CERFLPs|An SMOCP model . 146 7 Conclusion 150 viii List of Abbreviations CERFLP continuous Euclidean-rectilinear facility location problem CFLP continuous facility location problem DERFLP discrete Euclidean-rectilinear facility location problem DFLP discrete facility location problem DLP deterministic linear programming DRQCP deterministic rotated quadratic cone programming DSDP deterministic semidefinite programming DSOCP deterministic second-order cone programming DSP deterministic symmetric programming EFLP Euclidean facility location problem ERFLP Euclidean-rectilinear facility location problem ESFLP Euclidean single facility location problem FLP facility location problem KKT Karush-Kuhn-Tucker conditions MFLP multiple facility location problem MIMOCP mixed integer multi-order cone programming MOCP (deterministic) multi-order cone programming 0 − 1MOCP 0-1 multi-order cone programming POCP pth − order cone programming RFLP rectilinear facility location problem SFLP stochastic facility location problem SLP stochastic linear programming SMOCP stochastic multi-order cone programming SRQCP stochastic rotated quadratic cone programming SSDP stochastic semidefinite programming ix SSOCP stochastic second-order cone programming SSP stochastic symmetric programming Arw(x) the arrow-shaped matrix associated with the vector x Aut(K) the automorphism group of a cone K diag(·) the operator that maps its argument to a block diagonal matrix GL(n; R) the general linear group of degree n over R int(K) the interior of a cone K E n the n dimensional real vector space whose elements are indexed from 0 n E+ the second-order cone of dimension n ^n E+ the rotated quadratic cone of dimension n Hn the space of complex Hermitian matrices of order n n H+ the space of complex Hermitian semidefinite matrices of order n KJ the cone of squares of a Euclidean Jordan algebra J n th Qp the p -order cone of dimension n QHn the space of quaternion Hermitian matrices of order n n QH+ the space of quaternion Hermitian semidefinite matrices of order n Sn the space of real symmetric matrices of order n n S+ the space of real symmetric positive semidefinite matrices of order n Rn the space of real vectors of dimension n n n R+ the cone of nonnegative orthants of R jjxjj the Frobenius norm of an element x jjxjj2 the Euclidean norm of a vector x jjxjjp the p-norm of a vector x bxc the linear representation of an element x dxe the quadratic representation of an element x x X 0 the matrix X is positive semidefinite X 0 the matrix X is positive definite x 0 the vector x lies in a second-order cone of an appropriate dimension x 0 the vector x lies in the interior of a second-order cone of an appropriate dimension x N 0 the vector x lies in the Cartesian product of N second-order cones with appropriate dimensions x ^ 0 the vector x lies in a rotated quadratic cone of an appropriate dimension x ^ N 0 the vector x lies in the Cartesian product of N rotated quadratic cones with appropriate dimensions x KJ 0 x is an element of a symmetric cone KJ x KJ 0 x is an element of the interior of a symmetric cone KJ th x hpi 0 the vector x lies in a p -order cone of an appropriate dimension th x Nhpi 0 the vector x lies in the Cartesian product of N p -order cones with appropriate dimensions x hp1;p2;:::;pN i 0 the vector x lies in the Cartesian product of N cones of orders p1; p2;:::; pN and with appropriate dimensions x ◦ y the Jordan multiplication of elements x and y of a Jordan algebra x • y the inner product trace(x ◦ y) of elements x and y of a Euclidean Jordan algebra xi Chapter 1 Introduction and Background 1.1 Introduction The purpose of this dissertation is to introduce the two-stage stochastic symmetric pro- grams (SSPs)1 with recourse and to study this problem in the dual standard form: max cTx + E [Q(x;!)] s.t.