A convex optimization problem in conic linear form is an optimization problem of the form CP (d) : maximize c T

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This paper presents a unified study of duality properties for the problem of minimizing a linear function over the intersection of an affine space with a convex cone infinite dimension. Existing duality results are carefully surveyed and some new duality properties are established. Examples are given to illustrate these new properties. The topics covered in this paper include Gordon-Stiemke type theorems, Farkas type theorems, perfect duality, Slater condition, regularization, Ramana's duality, and approximate dualities. The dual representations of various convex sets, convex cones and conic convex programs are also discussed.

We consider the Riemannian geometry defined on a convex set by the Hessian of a selfconcordant barrier function, and its associated geodesic curves. These provide guidance for the construction of efficient interior-point methods for optimizing a linear function over the intersection of the set with an affine manifold. We show that algorithms that follow the primal-dual central path are in some sense close to optimal. The same is true for methods that follow the shifted primal-dual central path among certain infeasible-interior-point methods. We also compute the geodesics in several simple sets. ∗ Copyright (C) by Springer-Verlag. Foundations of Computational Mathewmatics 2 (2002), 333–361.

How to initialize an algorithm to solve an optimization problem is of great theoretical and practical importance. In the simplex method for linear programming this issue is resolved by either the two-phase approach or using the so-called big M technique. In the interior point method, there is a more elegant way to deal with the initialization problem, viz. the self-dual embedding technique proposed by Ye, Todd and Mizuno [30]. For linear programming this technique makes it possible to identify an optimal solution or conclude the problem to be infeasible/unbounded by solving its embedded self-dual problem. The embedded self-dual problem has a trivial initial solution and has the same structure as the original problem. Hence, it eliminates the need to consider the initialization problem at all. In this paper, we extend this approach to solve general conic convex programming, including semidefinite programming. Since a nonlinear conic convex programming problem may lack the so-called stri...

In this paper we introduce a conic optimization formulation for inequality-constrained convex programming, and propose a self-dual embedding model for solving the resulting conic optimization problem. The primal and dual cones in this formulation are characterized by the original constraint functions and their corresponding conjugate functions respectively. Hence they are completely symmetric. This allows for a standard primal-dual path following approach for solving the embedded problem. Moreover, there are two immediate logarithmic barrier functions for the primal and dual cones. We show that these two logarithmic barrier functions are conjugate to each other. The explicit form of the conjugate functions are in fact not required to be known in the algorithm. An advantage of the new approach is that there is no need to assume an initial feasible solution to start with. To guarantee the polynomiality of the path-following procedure, we may apply the self-concordant barrier theory of Nesterov and Nemirovski. For this purpose, as one application, we prove that the barrier functions constructed this way are indeed self-concordant when the original constraint functions are convex and quadratic. Keywords: Convex Programming, Convex Cones, Self-Dual Embedding, Self-Concordant Barrier Functions. # Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong. Research supported by Hong Kong RGC Earmarked Grants CUHK4181/00E and CUHK4233/01E. 1 1

We consider the problem of optimizing a linear function over the intersection of an a#ne space and a special class of closed, convex cones, namely the symmetric cones over the reals. This problem subsumes linear programming, convex quadratically constrained quadratic programming, and semidefinite programming as special cases. First, we derive some perturbation results for this problem class. Then, we discuss two solution methods: an interior-- point method capable of delivering highly accurate solutions to problems of modest size, and a first order bundle method which provides solutions of low accuracy, but can handle much larger problems. Finally, we describe an application of semidefinite programming in electronic structure calculations, and give some numerical results on sample problems. vi Contents Dedication iii Acknowledgment iv Abstract vi List of Figures ix List of Tables x List of Symbols and Notations x 1 Conic Optimization Problems 1 1.1 Problem Formulation . . . . . . . ...

In this paper we present a general theory for transforming a normalized homogeneous conic system F: Ax = 0, ¯s T x = 1, x ∈ C to an equivalent system via projective transformation induced by the choice of a point ˆv in the set H ◦ ¯s = {v: ¯s − AT v ∈ C ∗}. Such a projective transformation serves to pre-condition the conic system into a system that has both geometric and computational properties with certain guarantees. We characterize both the geometric behavior and the computational behavior of the transformed system as a function of the symmetry of ˆv in H ◦ ¯s as well as the complexity parameter ϑ of the barrier for C. Under the assumption that F has an interior solution, H ◦ ¯s must contain a point v whose symmetry is at least 1/m; if we can find a point whose symmetry is Ω(1/m) then we can projectively transform the conic system to one whose geometric properties and computational complexity will be strongly-polynomial-time in m and ϑ. We present a method for generating such a point ˆv based on sampling and on a geometric random walk on H ◦ ¯s with associated complexity and probabilistic analysis. Finally, we implement this methodology on randomly generated homogeneous linear programming feasibility problems, constructed to be poorly behaved. Our computational results indicate that the projective pre-conditioning methodology holds the promise to markedly reduce the overall computation time for conic feasibility problems; for instance we observe a 46 % decrease in average IPM iterations for 100 randomly generated poorly-behaved problem instances of dimension 1000 × 5000.

We present a primal-dual infeasible interior-point algorithm. As usual, the algorithm decreases the duality gap and the feasibility residuals at the same rate. Assuming that an optimal solution exists it is shown that at most O(n) iterations suffice to reduce the duality gap and the residuals by the factor 1/e. This implies an O(nlog(n/ε)) iteration bound for getting an ε-solution of the problem at hand, which coincides with the best known bound for infeasible interior-point algorithms. The algorithm constructs strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem. A special feature of the algorithm is that it uses only full-Newton steps. Two types of full-Newton steps are used, so-called feasibility steps and usual (centering) steps. Starting at strictly feasible iterates of a perturbed pair, (very) close its central path, feasibility steps serve to generate strictly feasible iterates for the next perturbed pair. By accomplishing a few centering steps for the new perturbed pair we obtain strictly feasible iterates close enough to the central path of the new perturbed pair. The algorithm finds an optimal solution or detects infeasibility or unboundedness of the given problem.

In this paper we study the homogeneous conic system F: Ax =0, x ∈ C \{0}. We choose a point ¯s ∈ intC ∗ that serves as a normalizer and consider computational properties of the normalized system F¯s: Ax = 0, ¯s T x =1, x ∈ C. We show that the computational complexity of solving F via an interior-point method depends only on the complexity value ϑ of the barrier for C and on the symmetry of the origin in the image set H¯s: = {Ax: ¯s T x =1, x ∈ C}, where the symmetry of 0 in H¯s is sym(0,H¯s):=max{α: y ∈ H¯s ⇒−αy ∈ H¯s}. We show that a solution of F can be computed in O ( √ ϑ ln(ϑ/sym(0,H¯s)) interior-point iterations. In order to improve the theoretical and practical computation of a solution of F, we next present a general theory for projective re-normalization of the feasible region F¯s and the image set H¯s and prove the existence of a normalizer ¯s such that sym(0,H¯s) ≥ 1/m provided that F has an interior solution. We develop a methodology for constructing a normalizer ¯s such that sym(0,H¯s) ≥ 1/m with high probability, based on sampling on a geometric random walk with associated probabilistic complexity analysis. While such a normalizer is not itself computable in strongly-polynomialtime, the normalizer will yield a conic system that is solvable in O ( √ ϑ ln(mϑ)) iterations, which is strongly-polynomialtime. Finally, we implement this methodology on randomly generated homogeneous linear programming feasibility problems, constructed to be poorly behaved. Our computational results indicate that the projective re-normalization methodology holds the promise to markedly reduce the overall computation time for conic feasibility problems; for instance we observe a 46 % decrease in average IPM iterations for 100 randomly generated poorly-behaved problem instances of dimension 1000 × 5000.