Complexity analysis of primal-dual algorithms for the semidefinite linear complementarity problem

Complexity analysis and numerical implementation of a short-step primal-dual algorithm for linear complementarity problems

In this paper we propose a new class of primal-dual path-following interior point algorithms for solving monotone linear complementarity problems. At each iteration, the method would select a target on the central path with a large update from the current iterate, and then the Newton method is used to get the search directions, followed by adaptively choosing the step sizes, which are, e.g., the largest possible steps before leaving a neighborhood that is as wide as the given ${\cal N}^-_{\infty}$ neighborhood. The only deviation from the classical approach is that we treat the classical Newton direction as the sum of two other directions, corresponding to, respectively, the negative part and the positive part of the right-hand side. We show that if these two directions are equipped with different and appropriate step sizes, then the method enjoys the low iteration bound of $O(\sqrt{n}\log L)$, where n is the dimension of the problem and $L=\frac{(x^0)^Ts^0}{\ep}$ with $\ep$ the required precision and $(x^0,s^0)$ the initial interior solution. For a predictor-corrector variant of the method, we further prove that, besides the predictor steps, each corrector step also reduces the duality gap by a rate of $1-1/O(\sqrt{n})$. Additionally, if the problem has a strict complementary solution, then the predictor steps converge Q-quadratically.

Kernel functions serve the central goal of creating new search directions for the primal-dual interiorpoint algorithm to solve linear optimization problems. A significantly improved primal-dual interior-point algorithm for linear optimization is presented based on a novel kernel function. We show a primal-dual interior-point technique for linear optimization based on a class of kernel functions that are eligible. This research presents a new efficient kernel function-based primal-dual IPM algorithm for semidefinite programming problems based on the Nesterov-Todd (NT) direction. With a new and simple technique, we propose a new kernel function to obtain an optimal solution of the perturbed problem (SDP)µ. We obtain the best-known complexity results,for smalland large-update namely O(pp+1/2p √nlog tr(X0S0)/ε) and O((pn)p+1/2p log trX0S0/ε) large update To prove the effectiveness of our proposed kernel function, we compare our numerical results with some alternatives presented by Touil et al. (2017).

Circular programming problems are a new class of convex optimization problems that include second-order cone programming problems as a special case‎. ‎Alizadeh and Goldfarb [Math‎. ‎Program‎. ‎Ser‎. ‎A 95 (2003) 3--51] introduced primal-dual path-following algorithms for solving second-order cone programming problems‎. ‎In this paper‎, ‎we generalize their work by using the machinery of Euclidean Jordan algebras associated with the circular cones to derive primal-dual path-following interior point algorithms for circular programming problems‎. ‎We prove polynomial convergence of the proposed algorithms by showing that the circular logarithmic barrier is a strongly self-concordant barrier‎. ‎The numerical examples show the path-following algorithms are simple and efficient‎.

Characterization of [formula omitted]-property for multiplicative transformations in semidefinite linear complementarity problems

Given a linear transformation L on a finite dimensional real inner product space V to itself and an element q ∈ V we consider the general linear complementarity problem LCP (L, K, q) on a proper cone K ⊆ V. We observe that the iterates generated by any closed algorithmic map will converge to a solution for LCP (L, K, q), whenever L is strongly monotone. Lipschitz constants of L is vital in establishing the above said convergence. Hence we compute the Lipschitz constants for certain classes of Lyapunov, Stein and double-sided multiplicative transformations in the setting of semidefinite linear complementarity problems. We give a numerical illustration of a closed algorithmic map in the setting of a standard linear complementarity problem. On account of the difficulties in numerically implementing such algorithms for general linear complementarity problems, we give an alternative algorithm for computing the solution for a special class of strongly monotone semidefinite linear complementarity problems along with a numerical example.

A full-Newton step [formula omitted] infeasible-interior-point algorithm for linear complementarity problems

We consider semidefinite monotone linear complementarity problems (SDLCP) in the space *** equation here *** n of real symmetric n×n-matrices equipped with the cone *** equation here *** n+ of all symmetric positive semidefinite matrices. One may define weighted (using any M∈ *** equation here *** n++ as weight) infeasible interior point paths by replacing the standard condition XY=rI, r>0, (that defines the usual central path) by (XY+YX)/2=rM. Under some mild assumptions (the most stringent is the existence of some strictly complementary solution of (SDLCP)), these paths have a limit as r↓0, and they depend analytically on all path parameters (such as r and M), even at the limit point r=0.

Kernel functions play an important role in defining new search directions for primal-dual interior-point algorithm. In this paper, a new kernel function which its barrier term is integral type is proposed. We study the properties of the new kernel function, and give a primal-dual interior-point algorithm for solving linear optimization based on the new kernel function. Polynomial complexity of algorithm is analyzed. The iteration bounds both for large-update and for small-update methods are obtained, respectively. The iteration bound for small-update method is the best known complexity bound.

We present a unified analysis for a class of long-step primal-dual path-following algorithms for semidefinite programming whose search directions are obtained through linearization of the symmetrized equation of the central path Hp(XS) -- [PXSP -~ + (PXSP 1)TI/2 = #I, introduced by Zhang. At an iterate (X, S), we choose a scaling matrix P from the class of nonsingular matrices P such that PXSP -~ is symmetric. This class of matrices includes the three well-known choices, namely: P = S ~/2 and P - X -~/2 proposed by Monteiro, and the matrix P corresponding to the Nesterov Todd direction. We show that within the class of algorithms studied in this paper, the one based on the Nesterov-Todd direction has the lowest possible iteration-complexity bound that can provably be derived from our analysis. More specifically, its iteration-complexity bound is of the same order as that of the corresponding long-step primal-dual path-following algorithm for linear programming introduced by Kojima, Mizuno and Yoshise. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

In this paper we propose a class of new large-update primal-dual interior-point algorithms for P *(κ) nonlinear complementarity problem (NCP), which are based on a class of kernel functions investigated by Bai et al. in their recent work for linear optimization (LO). The arguments for the algorithms are followed as Peng et al.’s for P *(κ) complementarity problem based on the self-regular functions [Peng, J., Roos, C., Terlaky, T.: Self-Regularity: A New Paradigm for Primal-Dual Interior-Point Algorithms, Princeton University Press, Princeton, 2002]. It is worth mentioning that since this class of kernel functions includes a class of non-self-regular functions as special case, so our algorithms are different from Peng et al.’s and the corresponding analysis is simpler than theirs. The ultimate goal of the paper is to show that the algorithms based on these functions have favorable polynomial complexity.

Based on a new parametric kernel function, this paper presents a primal-dual large-update interior-point algorithm (IPM) for semi-definite optimization (SDO) problems. The new parametric function is neither self-regular function nor the usual logarithmic barrier function. It is strongly convex and possesses some novel analytic properties. We analyse this new parametric kernel function and show that the proposed algorithm has favorable complexity bound in terms of the analytic properties of the kernel function. Moreover, the complexity bound for our large-update IPM is shown to be O(\sqrt{n}(\log n)^2 \log\frac{n}{\epsilon}). Some numerical results are reported to illustrate the feasibility of the proposed algorithm.

In the present paper we present a class of polynomial primal-dual interior-point algorithms for semidefinite optimization based on a kernel function. This kernel function is not a so-called self-regular function due to its growth term increasing linearly. Some new analysis tools were developed which can be used to deal with complexity analysis of the algorithms which use analogous strategy in [5] to design the search directions for the Newton system. The complexity bounds for the algorithms with large- and small-update methods were obtained, namely, O(qn(p+q)/q(p+1)) log n/ɛ and O(q2 √n) log n/ɛ, respectively.

A class of polynomial primal-dual interior-point algorithms for P * (κ ) linear complementarity problems (LCPs) are presented. We generalize Ghami et al.'s[A polynomial-time algorithm for linear optimization based on a new class of kernel functions(2008)] algorithm for linear optimization (LO) problem to P * (κ ) LCPs. Our analysis is based on a class of finite kernel functions which have the linear and quadratic growth terms. Since P *(κ ) LCP is a generalization of LO problem, we lose the orthogonality of the vectors dx and ds . So our analysis is different from the one in Ghami et al's algorithm. Despite this, the favorable complexity result is obtained, namely, $O((1 + 2\kappa)n^{\frac{1}{1+p}}\log n \log(n/\epsilon))$, which is better than the usual large-update primal-dual algorithm based on the classical logarithmic barrier function for P * (κ ) LCP.

An interior point method (IPM) defines a search direction at an interior point of the feasible region. These search directions form a direction field, which in turn defines a system of ordinary differential equations (ODEs). The solutions of the system of ODEs are called off-central paths, underlying paths lying in the interior of the feasible region. It is known that not all off-central paths are analytic, whether w.r.t. μ or \(\sqrt{\mu}\) , where μ represents the duality gap, at a solution of a given semidefinite linear complementarity problem, SDLCP (Sim and Zhao, Math. Program. 110:475–499, 2007). In Sim and Zhao (J. Optim. Theory Appl. 137:11–25, 2008), we give a necessary and sufficient condition for when an off-central path is analytic as a function of \(\sqrt{\mu}\) at a solution of a general SDLCP. It is then natural to ask about the analyticity of a SDLCP off-central path at a solution, as a function of μ. We investigate this in the current paper. Again, we work under the assumption that the given SDLCP satisfies strict complementarity condition.