Polynomial Time Interior Point Methods Research Articles

Set-Limited Functions and Polynomial-Time Interior-Point Methods

AbstractIn this paper, we revisit some elements of the theory of self-concordant functions. We replace the notion of self-concordant barrier by a new notion of set-limited function, which forms a wider class. We show that the proper set-limited functions ensure polynomial time complexity of the corresponding path-following method (PFM). Our new PFM follows a deviated path, which connects an arbitrary feasible point with the solution of the problem. We present some applications of our approach to the problems of unconstrained optimization, for which it ensures a global linear rate of convergence even in for nonsmooth objective function.

Local Superlinear Convergence of Polynomial-Time Interior-Point Methods for Hyperbolicity Cone Optimization Problems

In this paper, we establish the local superlinear convergence property of some polynomial-time interior-point methods for an important family of conic optimization problems. The main structural property used in our analysis is the logarithmic homogeneity of self-concordant barrier function, which must have negative curvature. We propose a new path-following predictor-corrector scheme, which works only in the dual space. It is based on an easily computable gradient proximity measure, which ensures an automatic transformation of the global linear rate of convergence to the local superlinear one under some mild assumptions. Our step-size procedure for the predictor step is related to the maximum step size maintaining feasibility. As the optimal solution set is approached, our algorithm automatically tightens the neighborhood of the central path proportionally to the current duality gap.

Hessian distances and their applications in the complexity analysis of interior-point methods

For interior points in convex cones, we introduce the Hessian distance function, and show that it is convenient for complexity analysis of polynomial-time interior-point methods (IPMs). As an example of its application, we develop new infeasible-start IPM for the linear conic optimization problem. In our setting, the primal and dual cones need not to be self-dual. We can start from any primal–dual point in the interior of the cones. Then, the damped Newton's method can be used for obtaining an approximate solution for the strictly feasible case, or for detecting primal/dual infeasibility. The complexity of these cases depends on the Hessian distance between the starting point and the feasibility/infeasibility certificates. We also present some numerical results.

Notes on Computational Complexity of GE Inequalities

This paper is a revision of my paper, CFDP 1865. The principal innovation is an equivalent reformulation of the decision problem for weak feasibility of the GE inequalities, using polynomial time ellipsoid methods, as a semidefinite optimization problem, using polynomial time interior point methods. We minimize the maximum of the Euclidean distances between the aggregate endowment and the Minkowski sum of the sets of consumer's Marshallian demands in each observation. We show that this is an instance of the generic semidefinite optimization problem: inf_{x in K}f(x) = Opt(K,f), the optimal value of the program,where the convex feasible set K and the convex objective function f(x) have semidefinite representations. This problem can be approximately solved in polynomial time. That is, if p(K,x) is a convex measure of infeasibilty, where for all x, p(K,x) > 0 and p(K,z) = 0 iff z in K, then for every epsilon > 0 there exists an epsilon-optimal y such that p(K,y)

An interior-point algorithm for solving inverse linear optimization problem

In this article, we deal with the inverse linear optimization problem under ℓ2-norm defined as follows: for a given linear optimization problem LP(b), with the right-hand side vector b, and current operating plan x 0, we would like to perturb the right-hand side vector b to so that x 0 is an optimal solution of and is minimum. We prove that this inverse problem is equivalent to an optimization problem which contains some equilibrium constraints that makes it hard to solve. In order to solve this optimization problem, we employ the interior point approach behind LOQO and show that, at each iteration, the search direction is a descent direction for the ℓ2 merit function. Therefore, an ε-solution of the inverse problem can be computed using an extension of polynomial-time interior-point methods for linear and quadratic optimization problems. Finally, computational results of applying proposed approach on some generated inverse problems are given.

Notes on Computational Complexity of GE Inequalities

This paper is a revision of my paper, CFDP 1865. The principal innovation is an equivalent reformulation of the decision problem for weak feasibility of the GE inequalities, using polynomial time ellipsoid methods, as a semidefinite optimization problem, using polynomial time interior point methods. We minimize the maximum of the Euclidean distances between the aggregate endowment and the Minkowski sum of the sets of consumer's Marshallian demands in each observation. We show that this is an instance of the generic semidefinite optimization problem: inf_{x in K}f(x) equivalent to Opt(K,f), the optimal value of the program, where the convex feasible set K and the convex objective function f(x) have semidefinite representations. This problem can be approximately solved in polynomial time. That is, if p(K,x) is a convex measure of infeasibilty, where for all x, p(K,x) >= 0 and p(K,z) = 0 iff z in K, then for every epsilon > 0 there exists an epsilon-optimal y such that p(K,y)

A kernel function based Interior-Point Methods for solving P *(κ)-linear complementarity problem

In this paper, motivated by the complexity results of Interior Point Methods (IPMs) for Linear Optimization (LO) based on kernel functions, we present a polynomial time IPM for solving P*(κ)-linear complementarity problem, using a new class of kernel functions. The special case of our new class was considered earlier for LO by Y. Q. Bai et al. in 2004. Using some appealing properties of the new class, we show that the iteration bound for IPMs matches the so far best known theoretical iteration bound for both large and small updates by choosing special values for the parameters of the new class.

On a Class of Superlinearly Convergent Polynomial Time Interior Point Methods for Sufficient LCP

A new class of infeasible interior point methods for solving sufficient linear complementarity problems (LCPs) requiring one matrix factorization and $m$ backsolves at each iteration is proposed and analyzed. The algorithms from this class use a large $({\cal N}_\infty^-$) neighborhood of an infeasible central path associated with the complementarity problem and an initial positive, but not necessarily feasible, starting point. The Q-order of convergence of the complementarity gap, the residual, and the iteration sequence is $m+1$ for problems that admit a strict complementarity solution and $(m+1)/2$ for general sufficient LCPs. The methods do not depend on the handicap $\kappa$ of the sufficient LCP. If the starting point is feasible (or “almost” feasible), the proposed algorithms have ${\cal O}((1+\kappa)(1+\log\sqrt[m]{1+\kappa}\,)\sqrt{n}\;L)$ iteration complexity, while if the starting point is “large enough,” the iteration complexity is ${\cal O}((1+\kappa)^{2+1/m}(1+\log\sqrt[m]{1+\kappa}\,)n\;L)$.

Computing closest saddle node bifurcations in a radial system via conic programming

This paper considers the problem of computing the loading limits in a radial system which are (i) locally closest to current operating load powers and (ii) at which saddle node bifurcation occurs. The procedure is based on a known technique which requires iterating between two computational steps until convergence. In essence, step 1 produces a vector normal to the real and/or reactive load solution space boundary, whereas step 2 computes the bifurcation point along that vector. The paper shows that each of the above computational steps can be formulated as a second-order cone program for which polynomial time interior-point methods and efficient implementations exist. The proposed conic programming approach is used to compute the closest bifurcation points and the corresponding worst case load power margins of eleven different distribution systems. The approach is validated graphically and the existence of multiple load power margins is investigated.

A Conic Quadratic Format for the Load Flow Equations of Meshed Networks

Recent research has shown that the load flow equations describing the steady-state conditions in a radial network can be formulated as a second-order cone program and solved using polynomial-time interior point methods. This paper extends the conic formulation to meshed networks by including a trigonometric functional constraint for the voltage angle spread on each line in the network. In particular, it is shown that the load flow solution of a meshed network can be obtained from a sequence of conic programming solutions. The proposed formulation and iterative procedure are validated by comparing with the Newton-Raphson solution of the standard power flow model. Testing is reported on seven networks. The proposed format of equations opens the door for new implementations of energy management system functions based on efficient and numerically robust conic quadratic optimization methods.

Optimal Design of Nearfield Wideband Beamformers Robust Against Errors in Microphone Array Characteristics

Nearfield wideband beamformers for microphone arrays have wide applications, such as hands-free telephony, hearing aids, and speech input devices to computers. The existing design approaches for nearfield wideband beamformers are highly sensitive to errors in microphone array characteristics, i.e., microphone gain, phase, and position errors, as well as sound speed errors. In this paper, a robust design approach for nearfield wideband beamformers for microphone arrays is proposed. The robust nearfield wideband beamformers are designed based on the minimax criterion with the worst case performance optimization. The design problems can be formulated as second-order cone programming and be solved efficiently using the well-established polynomial time interior-point methods. Several interesting properties of the robust nearfield wideband beamformers are derived. Numerical examples are given to demonstrate the efficacy of the proposed beamformers in the presence of errors in microphone array characteristics.

?Cone-free? primal-dual path-following and potential-reduction polynomial time interior-point methods

We present a framework for designing and analyzing primal-dual interior-point methods for convex optimization. We assume that a self-concordant barrier for the convex domain of interest and the Legendre transformation of the barrier are both available to us. We directly apply the theory and techniques of interior-point methods to the given good formulation of the problem (as is, without a conic reformulation) using the very usual primal central path concept and a less usual version of a dual path concept. We show that many of the advantages of the primal-dual interior-point techniques are available to us in this framework and therefore, they are not intrinsically tied to the conic reformulation and the logarithmic homogeneity of the underlying barrier function.

Robust Convex Optimization

We study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set U, yet the constraints must hold for all possible values of the data from U. The ensuing optimization problem is called robust optimization. In this paper we lay the foundation of robust convex optimization. In the main part of the paper we show that if U is an ellipsoidal uncertainty set, then for some of the most important generic convex optimization problems (linear programming, quadratically constrained programming, semidefinite programming and others) the corresponding robust convex program is either exactly, or approximately, a tractable problem which lends itself to efficient algorithms such as polynomial time interior point methods.