Polynomial convergence of primal-dual algorithms for the second-order cone program based on the MZ-family of directions

In this paper we study primal-dual path-following algorithms for the second-order cone programming (SOCP) based on a family of directions that is a natural extension of the Monteiro-Zhang (MZ) family for semidefinite programming. We show that the polynomial iteration-complexity bounds of two well-known algorithms for linear programming, namely the short-step path-following algorithm of Kojima et al. and Monteiro and Adler, and the predictor-corrector algorithm of Mizuno et al., carry over to the context of SOCP, that is they have an O( logε^-1) iteration-complexity to reduce the duality gap by a factor of ε, where n is the number of second-order cones. Since the MZ-type family studied in this paper includes an analogue of the Alizadeh, Haeberly and Overton pure Newton direction, we establish for the first time the polynomial convergence of primal-dual algorithms for SOCP based on this search direction. Received: June 5, 1998 / Accepted: September 8, 1999?Published online April 20, 2000     

Self-regular functions and new search directions for linear and semidefinite optimization

In this paper, we introduce the notion of a self-regular function. Such a function is strongly convex and smooth coercive on its domain, the positive real axis. We show that any such function induces a so-called self-regular proximity function and a corresponding search direction for primal-dual path-following interior-point methods (IPMs) for solving linear optimization (LO) problems. It is proved that the new large-update IPMs enjoy a polynomial ?(n log) iteration bound, where q≥1 is the so-called barrier degree of the kernel function underlying the algorithm. The constant hidden in the ?-symbol depends on q and the growth degree p≥1 of the kernel function. When choosing the kernel function appropriately the new large-update IPMs have a polynomial ?(lognlog) iteration bound, thus improving the currently best known bound for large-update methods by almost a factor . Our unified analysis provides also the ?(log) best known iteration bound of small-update IPMs. At each iteration, we need to solve only one linear system. An extension of the above results to semidefinite optimization (SDO) is also presented. Received: March 2000 / Accepted: December 2001?Published online April 12, 2002     

Convergence and efficiency of subgradient methods for quasiconvex minimization

We study a general subgradient projection method for minimizing a quasiconvex objective subject to a convex set constraint in a Hilbert space. Our setting is very general: the objective is only upper semicontinuous on its domain, which need not be open, and various subdifferentials may be used. We extend previous results by proving convergence in objective values and to the generalized solution set for classical stepsizes t _k →0, ∑t _k =∞, and weak or strong convergence of the iterates to a solution for {t _k }∈ℓ₂∖ℓ₁ under mild regularity conditions. For bounded constraint sets and suitable stepsizes, the method finds ε-solutions with an efficiency estimate of O(ε^-2), thus being optimal in the sense of Nemirovskii. Received: October 4, 1998 / Accepted: July 24, 2000?Published online January 17, 2001     

A superlinear space decomposition algorithm for constrained nonsmooth convex program

A class of constrained nonsmooth convex optimization problems, that is, piecewise C² convex objectives with smooth convex inequality constraints are transformed into unconstrained nonsmooth convex programs with the help of exact penalty function. The objective functions of these unconstrained programs are particular cases of functions with primal-dual gradient structure which has connection with VU space decomposition. Then a VU space decomposition method for solving this unconstrained program is presented. This method is proved to converge with local superlinear rate under certain assumptions. An illustrative example is given to show how this method works.     

Discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization

Based on the authors’ previous work which established theoretical foundations of two, conceptual, successive convex relaxation methods, i.e., the SSDP (Successive Semidefinite Programming) Relaxation Method and the SSILP (Successive Semi-Infinite Linear Programming) Relaxation Method, this paper proposes their implementable variants for general quadratic optimization problems. These problems have a linear objective function c ^T x to be maximized over a nonconvex compact feasible region F described by a finite number of quadratic inequalities. We introduce two new techniques, “discretization” and “localization,” into the SSDP and SSILP Relaxation Methods. The discretization technique makes it possible to approximate an infinite number of semi-infinite SDPs (or semi-infinite LPs) which appeared at each iteration of the original methods by a finite number of standard SDPs (or standard LPs) with a finite number of linear inequality constraints. We establish:?•Given any open convex set U containing F, there is an implementable discretization of the SSDP (or SSILP) Relaxation Method which generates a compact convex set C such that F⊆C⊆U in a finite number of iterations.?The localization technique is for the cases where we are only interested in upper bounds on the optimal objective value (for a fixed objective function vector c) but not in a global approximation of the convex hull of F. This technique allows us to generate a convex relaxation of F that is accurate only in certain directions in a neighborhood of the objective direction c. This cuts off redundant work to make the convex relaxation accurate in unnecessary directions. We establish:?•Given any positive number ε, there is an implementable localization-discretization of the SSDP (or SSILP) Relaxation Method which generates an upper bound of the objective value within ε of its maximum in a finite number of iterations. Received: June 30, 1998 / Accepted: May 18, 2000?Published online September 20, 2000     

On convergence of the Gauss-Newton method for convex composite optimization

The local quadratic convergence of the Gauss-Newton method for convex composite optimization f=h∘F is established for any convex function h with the minima set C, extending Burke and Ferris’ results in the case when C is a set of weak sharp minima for h. Received: July 24, 1998 / Accepted: November 29, 2000?Published online September 3, 2001     

On the convexity of the multiplicative potential and penalty functions and related topics

It is well known that a function f of the real variable x is convex if and only if (x,y)→yf(y ^-1 x),y>0 is convex. This is used to derive a recursive proof of the convexity of the multiplicative potential function. In this paper, we obtain a conjugacy formula which gives rise, as a corollary, to a new rule for generating new convex functions from old ones. In particular, it allows to extend the aforementioned property to functions of the form (x,y)→g(y)f(g(y)^-1 x) and provides a new tool for the study of the multiplicative potential and penalty functions. Received: June 3, 1999 / Accepted: September 29, 2000?Published online January 17, 2001     

On error bounds for lower semicontinuous functions

We give some sufficient conditions for proper lower semicontinuous functions on metric spaces to have error bounds (with exponents). For a proper convex function f on a normed space X the existence of a local error bound implies that of a global error bound. If in addition X is a Banach space, then error bounds can be characterized by the subdifferential of f. In a reflexive Banach space X, we further obtain several sufficient and necessary conditions for the existence of error bounds in terms of the lower Dini derivative of f. Received: April 27, 2001 / Accepted: November 6, 2001?Published online April 12, 2002     

Convex composite non–Lipschitz programming

In this paper necessary, and sufficient optimality conditions are established without Lipschitz continuity for convex composite continuous optimization model problems subject to inequality constraints. Necessary conditions for the special case of the optimization model involving max-min constraints, which frequently arise in many engineering applications, are also given. Optimality conditions in the presence of Lipschitz continuity are routinely obtained using chain rule formulas of the Clarke generalized Jacobian which is a bounded set of matrices. However, the lack of derivative of a continuous map in the absence of Lipschitz continuity is often replaced by a locally unbounded generalized Jacobian map for which the standard form of the chain rule formulas fails to hold. In this paper we overcome this situation by constructing approximate Jacobians for the convex composite function involved in the model problem using ε-perturbations of the subdifferential of the convex function and the flexible generalized calculus of unbounded approximate Jacobians. Examples are discussed to illustrate the nature of the optimality conditions. Received: February 2001 / Accepted: September 2001?Published online February 14, 2002     

Non-special divisors on real algebraic curves and embeddings into real projective spaces

We show that there is a large class of non-special divisors of relatively small degree on a given real algebraic curve. If the real algebraic curve has many real components, such a divisor gives rise to an embedding (birational embedding, resp.) of the real algebraic curve into the real projective space ℙ ^r for r≥3 (r=2, resp.). We study these embeddings in quite some detail. Received: October 17, 2001?Published online: February 20, 2003     

Characterizations, bounds, and probabilistic analysis of two complexity measures for linear programming problems

This note studies _A , a condition number used in the linear programming algorithm of Vavasis and Ye [14] whose running time depends only on the constraint matrix A∈ℝ ^m×n , and (A), a variant of another condition number due to Ye [17] that also arises in complexity analyses of linear programming problems. We provide a new characterization of _A and relate _A and (A). Furthermore, we show that if A is a standard Gaussian matrix, then E(ln _A )=O(min{mlnn,n}). Thus, the expected running time of the Vavasis-Ye algorithm for linear programming problems is bounded by a polynomial in m and n for any right-hand side and objective coefficient vectors when A is randomly generated in this way. As a corollary of the close relation between _A and (A), we show that the same bound holds for E(ln(A)). Received: September 1998 / Accepted: September 2000?Published online January 17, 2001     

Logarithmic SUMT limits in convex programming

The limits of a class of primal and dual solution trajectories associated with the Sequential Unconstrained Minimization Technique (SUMT) are investigated for convex programming problems with non-unique optima. Logarithmic barrier terms are assumed. For linear programming problems, such limits – of both primal and dual trajectories – are strongly optimal, strictly complementary, and can be characterized as analytic centers of, loosely speaking, optimality regions. Examples are given, which show that those results do not hold in general for convex programming problems. If the latter are weakly analytic (Bank et al. [3]), primal trajectory limits can be characterized in analogy to the linear programming case and without assuming differentiability. That class of programming problems contains faithfully convex, linear, and convex quadratic programming problems as strict subsets. In the differential case, dual trajectory limits can be characterized similarly, albeit under different conditions, one of which suffices for strict complementarity. Received: November 13, 1997 / Accepted: February 17, 1999?Published online February 22, 2001     

A polynomial dual simplex algorithm for the generalized circulation problem

This paper presents a polynomial-time dual simplex algorithm for the generalized circulation problem. An efficient implementation of this algorithm is given that has a worst-case running time of O(m ²(m+nlogn)logB), where n is the number of nodes, m is the number of arcs and B is the largest integer used to represent the rational gain factors and integral capacities in the network. This running time is as fast as the running time of any combinatorial algorithm that has been proposed thus far for solving the generalized circulation problem. Received: June 1998 / Accepted: June 27, 2001?Published online September 17, 2001     

A descent method for submodular function minimization

We show a descent method for submodular function minimization based on an oracle for membership in base polyhedra. We assume that for any submodular function f: ?→R on a distributive lattice ?⊆2 ^V with ?,V∈? and f(?)=0 and for any vector x∈R ^V where V is a finite nonempty set, the membership oracle answers whether x belongs to the base polyhedron associated with f and that if the answer is NO, it also gives us a set Z∈? such that x(Z)>f(Z). Given a submodular function f, by invoking the membership oracle O(|V|²) times, the descent method finds a sequence of subsets Z ₁,Z ₂,···,Z _k of V such that f(Z ₁)>f(Z ₂)>···>f(Z _k )=min{f(Y) | Y∈?}, where k is O(|V|²). The method furnishes an alternative framework for submodular function minimization if combined with possible efficient membership algorithms. Received: September 9, 2001 / Accepted: October 15, 2001?Published online December 6, 2001     

A BFGS-IP algorithm for solving strongly convex optimization problems with feasibility enforced by an exact penalty approach

This paper introduces and analyses a new algorithm for minimizing a convex function subject to a finite number of convex inequality constraints. It is assumed that the Lagrangian of the problem is strongly convex. The algorithm combines interior point methods for dealing with the inequality constraints and quasi-Newton techniques for accelerating the convergence. Feasibility of the iterates is progressively enforced thanks to shift variables and an exact penalty approach. Global and q-superlinear convergence is obtained for a fixed penalty parameter; global convergence to the analytic center of the optimal set is ensured when the barrier parameter tends to zero, provided strict complementarity holds. Received: December 21, 2000 / Accepted: July 13, 2001?Published online February 14, 2002     

Mixing mixed-integer inequalities

Mixed-integer rounding (MIR) inequalities play a central role in the development of strong cutting planes for mixed-integer programs. In this paper, we investigate how known MIR inequalities can be combined in order to generate new strong valid inequalities.?Given a mixed-integer region S and a collection of valid “base” mixed-integer inequalities, we develop a procedure for generating new valid inequalities for S. The starting point of our procedure is to consider the MIR inequalities related with the base inequalities. For any subset of these MIR inequalities, we generate two new inequalities by combining or “mixing” them. We show that the new inequalities are strong in the sense that they fully describe the convex hull of a special mixed-integer region associated with the base inequalities.?We discuss how the mixing procedure can be used to obtain new classes of strong valid inequalities for various mixed-integer programming problems. In particular, we present examples for production planning, capacitated facility location, capacitated network design, and multiple knapsack problems. We also present preliminary computational results using the mixing procedure to tighten the formulation of some difficult integer programs. Finally we study some extensions of this mixing procedure. Received: April 1998 / Accepted: January 2001?Published online April 12, 2001     

A spectral bundle method with bounds

Semidefinite relaxations of quadratic 0-1 programming or graph partitioning problems are well known to be of high quality. However, solving them by primal-dual interior point methods can take much time even for problems of moderate size. The recent spectral bundle method of Helmberg and Rendl can solve quite efficiently large structured equality-constrained semidefinite programs if the trace of the primal matrix variable is fixed, as happens in many applications. We extend the method so that it can handle inequality constraints without seriously increasing computation time. In addition, we introduce inexact null steps. This abolishes the need of computing exact eigenvectors for subgradients, which brings along significant advantages in theory and in practice. Encouraging preliminary computational results are reported. Received: February 1, 2000 / Accepted: September 26, 2001?Published online August 27, 2002 RID="*" ID="*"A preliminary version of this paper appeared in the proceedings of IPCO ’98 [12].     

On the rank of mixed 0,1 polyhedra

For a polytope in the [0,1] ⁿ cube, Eisenbrand and Schulz showed recently that the maximum Chvátal rank is bounded above by O(n ²logn) and bounded below by (1+ε)n for some ε>0. Chvátal cuts are equivalent to Gomory fractional cuts, which are themselves dominated by Gomory mixed integer cuts. What do these upper and lower bounds become when the rank is defined relative to Gomory mixed integer cuts? An upper bound of n follows from existing results in the literature. In this note, we show that the lower bound is also equal to n. This result still holds for mixed 0,1 polyhedra with n binary variables. Received: March 15, 2001 / Accepted: July 18, 2001?Published online September 17, 2001     

A priori and universal estimates for global solutions of superlinear degenerate parabolic equations

We prove an a priori estimate and a universal bound for any global solution of the nonlinear degenerate reaction-diffusion equation u _t =Δu ^m +u ^p in a bounded domain with zero Dirichlet boundary conditions. Received: October 1, 2001?Published online: July 9, 2002     

Linear convergence of epsilon-subgradient descent methods for a class of convex functions

This paper establishes a linear convergence rate for a class of epsilon-subgradient descent methods for minimizing certain convex functions on ℝ ⁿ . Currently prominent methods belonging to this class include the resolvent (proximal point) method and the bundle method in proximal form (considered as a sequence of serious steps). Other methods, such as a variant of the proximal point method given by Correa and Lemaréchal, can also fit within this framework, depending on how they are implemented. The convex functions covered by the analysis are those whose conjugates have subdifferentials that are locally upper Lipschitzian at the origin, a property generalizing classical regularity conditions. Received March 29, 1996 / Revised version received March 5, 1999? Published online June 11, 1999