Concavity and efficient points of discrete distributions in probabilistic programming

We consider stochastic programming problems with probabilistic constraints involving integer-valued random variables. The concept of a p-efficient point of a probability distribution is used to derive various equivalent problem formulations. Next we introduce the concept of r-concave discrete probability distributions and analyse its relevance for problems under consideration. These notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables. The results are illustrated with numerical examples. Received: October 1998 / Accepted: June 2000?Published online October 18, 2000     

Upper Lipschitz behavior of solutions to perturbed. C1,1 programs

We analyze the local upper Lipschitz behavior of critical points, stationary solutions and local minimizers to parametric C ^1,1 programs. In particular, we derive a characterization of this property for the stationary solution set map without assuming the Mangasarian–Fromovitz CQ. Moreover, conditions which also ensure the persistence of solvability are given, and the special case of linear constraints is handled. The present paper takes pattern from [21] by continuing the approach via contingent derivatives of the Kojima function associated with the given optimization problem. Received: June 10, 1999 / Accepted: November 15, 1999?Published online July 20, 2000     

Error bounds for infinite systems of convex inequalities without Slater’s condition

The feasible set of a convex semi–infinite program is described by a possibly infinite system of convex inequality constraints. We want to obtain an upper bound for the distance of a given point from this set in terms of a constant multiplied by the value of the maximally violated constraint function in this point. Apart from this Lipschitz case we also consider error bounds of H?lder type, where the value of the residual of the constraints is raised to a certain power.?We give sufficient conditions for the validity of such bounds. Our conditions do not require that the Slater condition is valid. For the definition of our conditions, we consider the projections on enlarged sets corresponding to relaxed constraints. We present a condition in terms of projection multipliers, a condition in terms of Slater points and a condition in terms of descent directions. For the Lipschitz case, we give five equivalent characterizations of the validity of a global error bound.?We extend previous results in two directions: First, we consider infinite systems of inequalities instead of finite systems. The second point is that we do not assume that the Slater condition holds which has been required in almost all earlier papers. Received: April 12, 1999 / Accepted: April 5, 2000?Published online July 20, 2000     

Solving large quadratic assignment problems on computational grids

The quadratic assignment problem (QAP) is among the hardest combinatorial optimization problems. Some instances of size n = 30 have remained unsolved for decades. The solution of these problems requires both improvements in mathematical programming algorithms and the utilization of powerful computational platforms. In this article we describe a novel approach to solve QAPs using a state-of-the-art branch-and-bound algorithm running on a federation of geographically distributed resources known as a computational grid. Solution of QAPs of unprecedented complexity, including the nug30, kra30b, and tho30 instances, is reported. Received: September 29, 2000 / Accepted: June 5, 2001?Published online October 2, 2001     

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     

Error bounds for 2-regular mappings with Lipschitzian derivatives and their applications

We obtain local estimates of the distance to a set defined by equality constraints under assumptions which are weaker than those previously used in the literature. Specifically, we assume that the constraints mapping has a Lipschitzian derivative, and satisfies a certain 2-regularity condition at the point under consideration. This setting directly subsumes the classical regular case and the twice differentiable 2-regular case, for which error bounds are known, but it is significantly richer than either of these two cases. When applied to a certain equation-based reformulation of the nonlinear complementarity problem, our results yield an error bound under an assumption more general than b-regularity. The latter appears to be the weakest assumption under which a local error bound for complementarity problems was previously available. We also discuss an application of our results to the convergence rate analysis of the exterior penalty method for solving irregular problems. Received: February 2000 / Accepted: November 2000?Published online January 17, 2001     

Approximating the single source unsplittable min-cost flow problem

In the single source unsplittable min-cost flow problem, commodities must be routed simultaneously from a common source vertex to certain destination vertices in a given graph with edge capacities and costs; the demand of each commodity must be routed along a single path so that the total flow through any edge is at most its capacity. Moreover, the total cost must not exceed a given budget. This problem has been introduced by Kleinberg [7] and generalizes several NP-complete problems from various areas in combinatorial optimization such as packing, partitioning, scheduling, load balancing, and virtual-circuit routing. Kolliopoulos and Stein [9] and Dinitz, Garg, and Goemans [4] developed algorithms improving the first approximation results of Kleinberg for the problem of minimizing the violation of edge capacities and for other variants. However, known techniques do not seem to be capable of providing solutions without also violating the cost constraint. We give the first approximation results with hard cost constraints. Moreover, all our results dominate the best known bicriteria approximations. Finally, we provide results on the hardness of approximation for several variants of the problem. Received: August 23, 2000 / Accepted: April 20, 2001?Published online October 2, 2001     

A branch-and-cut method for 0-1 mixed convex programming

We generalize the disjunctive approach of Balas, Ceria, and Cornuéjols [2] and devevlop a branch-and-cut method for solving 0-1 convex programming problems. We show that cuts can be generated by solving a single convex program. We show how to construct regions similar to those of Sherali and Adams [20] and Lovász and Schrijver [12] for the convex case. Finally, we give some preliminary computational results for our method. Received January 16, 1996 / Revised version received April 23, 1999?Published online June 28, 1999     

A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities

In this paper we take a new look at smoothing Newton methods for solving the nonlinear complementarity problem (NCP) and the box constrained variational inequalities (BVI). Instead of using an infinite sequence of smoothing approximation functions, we use a single smoothing approximation function and Robinson’s normal equation to reformulate NCP and BVI as an equivalent nonsmooth equation H(u,x)=0, where H:ℜ ²ⁿ →ℜ ²ⁿ , u∈ℜ ⁿ is a parameter variable and x∈ℜ ⁿ is the original variable. The central idea of our smoothing Newton methods is that we construct a sequence {z ^k =(u ^k ,x ^k )} such that the mapping H(·) is continuously differentiable at each z ^k and may be non-differentiable at the limiting point of {z ^k }. We prove that three most often used Gabriel-Moré smoothing functions can generate strongly semismooth functions, which play a fundamental role in establishing superlinear and quadratic convergence of our new smoothing Newton methods. We do not require any function value of F or its derivative value outside the feasible region while at each step we only solve a linear system of equations and if we choose a certain smoothing function only a reduced form needs to be solved. Preliminary numerical results show that the proposed methods for particularly chosen smoothing functions are very promising. Received June 23, 1997 / Revised version received July 29, 1999?Published online December 15, 1999     

Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization

The strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in Euclidean space. These fundamental notions occur in various branches of convex optimization (constrained approximation, convex feasibility problems, linear inequalities, for instance). It is shown that the standard constraint qualification from convex analysis implies bounded linear regularity, which in turn yields the strong conical hull intersection property. Jameson’s duality for two cones, which relates bounded linear regularity to property (G), is re-derived and refined. For polyhedral cones, a statement dual to Hoffman’s error bound result is obtained. A sharpening of a result on error bounds for convex inequalities by Auslender and Crouzeix is presented. Finally, for two subspaces, property (G) is quantified by the angle between the subspaces. Received October 1, 1997 / Revised version received July 21, 1998? Published online June 11, 1999     

A bundle Bregman proximal method for convex nondifferentiable minimization

k } by taking x^k to be an approximate minimizer of , where is a piecewise linear model of f constructed from accumulated subgradient linearizations of f, D_h is the D-function of a generalized Bregman function h and t_k>0. Convergence under implementable criteria is established by extending our recent framework of Bregman proximal minimization, which is of independent interest, e.g., for nonquadratic multiplier methods for constrained minimization. In particular, we provide new insights into the convergence properties of bundle methods based on h=?|·|². Received September 18, 1997 / Revised version received June 30, 1998 Published online November 24, 1998     

On generalized trade-off directions in nonconvex multiobjective optimization

Trade-off information related to Pareto optimal solutions is important in multiobjective optimization problems with conflicting objectives. Recently, the concept of trade-off directions has been introduced for convex problems. These trade-offs are characterized with the help of tangent cones. Generalized trade-off directions for nonconvex problems can be defined by replacing convex tangent cones with nonconvex contingent cones. Here we study how the convex concepts and results can be generalized into a nonconvex case. Giving up convexity naturally means that we need local instead of global analysis. Received: December 2000 / Accepted: October 2001?Published online February 14, 2002     

An efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints

In this paper we investigate two approaches to minimizing a quadratic form subject to the intersection of finitely many ellipsoids. The first approach is the d.c. (difference of convex functions) optimization algorithm (abbr. DCA) whose main tools are the proximal point algorithm and/or the projection subgradient method in convex minimization. The second is a branch-and-bound scheme using Lagrangian duality for bounding and ellipsoidal bisection in branching. The DCA was first introduced by Pham Dinh in 1986 for a general d.c. program and later developed by our various work is a local method but, from a good starting point, it provides often a global solution. This motivates us to combine the DCA and our branch and bound algorithm in order to obtain a good initial point for the DCA and to prove the globality of the DCA. In both approaches we attempt to use the ellipsoidal constrained quadratic programs as the main subproblems. The idea is based upon the fact that these programs can be efficiently solved by some available (polynomial and nonpolynomial time) algorithms, among them the DCA with restarting procedure recently proposed by Pham Dinh and Le Thi has been shown to be the most robust and fast for large-scale problems. Several numerical experiments with dimension up to 200 are given which show the effectiveness and the robustness of the DCA and the combined DCA-branch-and-bound algorithm. Received: April 22, 1999 / Accepted: November 30, 1999?Published online February 23, 2000     

A geometric study of duality gaps, with applications

Lagrangian relaxation is often an efficient tool to solve (large-scale) optimization problems, even nonconvex. However it introduces a duality gap, which should be small for the method to be really efficient. Here we make a geometric study of the duality gap. Given a nonconvex problem, we formulate in a first part a convex problem having the same dual. This formulation involves a convexification in the product of the three spaces containing respectively the variables, the objective and the constraints. We apply our results to several relaxation schemes, especially one called “Lagrangean decomposition” in the combinatorial-optimization community, or “operator splitting” elsewhere. We also study a specific application, highly nonlinear: the unit-commitment problem. Received: June 1997 / Accepted: December 2000?Published online April 12, 2001     

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     

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].     

Nonlinear rescaling vs. smoothing technique in convex optimization

We introduce an alternative to the smoothing technique approach for constrained optimization. As it turns out for any given smoothing function there exists a modification with particular properties. We use the modification for Nonlinear Rescaling (NR) the constraints of a given constrained optimization problem into an equivalent set of constraints.?The constraints transformation is scaled by a vector of positive parameters. The Lagrangian for the equivalent problems is to the correspondent Smoothing Penalty functions as Augmented Lagrangian to the Classical Penalty function or MBFs to the Barrier Functions. Moreover the Lagrangians for the equivalent problems combine the best properties of Quadratic and Nonquadratic Augmented Lagrangians and at the same time are free from their main drawbacks.?Sequential unconstrained minimization of the Lagrangian for the equivalent problem in primal space followed by both Lagrange multipliers and scaling parameters update leads to a new class of NR multipliers methods, which are equivalent to the Interior Quadratic Prox methods for the dual problem.?We proved convergence and estimate the rate of convergence of the NR multipliers method under very mild assumptions on the input data. We also estimate the rate of convergence under various assumptions on the input data.?In particular, under the standard second order optimality conditions the NR method converges with Q-linear rate without unbounded increase of the scaling parameters, which correspond to the active constraints.?We also established global quadratic convergence of the NR methods for Linear Programming with unique dual solution.?We provide numerical results, which strongly support the theory. Received: September 2000 / Accepted: October 2001?Published online April 12, 2002     

Inverse functions of pseudo regular mappings and regularity conditions

We show that, even for monotone directionally differentiable Lipschitz functionals on Hilbert spaces, basic concepts of generalized derivatives identify only particular pseudo regular (or metrically regular) situations. Thus, pseudo regularity of (multi-) functions will be investigated by other means, namely in terms of the possible inverse functions. In this way, we show how pseudo regularity for the intersection of multifunctions can be directly characterized and estimated under general settings and how contingent and coderivatives may be modified to obtain sharper regularity conditions. Consequences for a concept of stationary points as limits of Ekeland points in nonsmooth optimization will be studied, too. Received: May 20, 1999 / Accepted: February 15, 2000?Published online July 20, 2000     

Global optimization of nonconvex factorable programming problems

In this paper, we consider a special class of nonconvex programming problems for which the objective function and constraints are defined in terms of general nonconvex factorable functions. We propose a branch-and-bound approach based on linear programming relaxations generated through various approximation schemes that utilize, for example, the Mean-Value Theorem and Chebyshev interpolation polynomials coordinated with a Reformulation-Linearization Technique (RLT). A suitable partitioning process is proposed that induces convergence to a global optimum. The algorithm has been implemented in C++ and some preliminary computational results are reported on a set of fifteen engineering process control and design test problems from various sources in the literature. The results indicate that the proposed procedure generates tight relaxations, even via the initial node linear program itself. Furthermore, for nine of these fifteen problems, the application of a local search method that is initialized at the LP relaxation solution produced the actual global optimum at the initial node of the enumeration tree. Moreover, for two test cases, the global optimum found improves upon the solutions previously reported in the source literature. Received: January 14, 1998 / Accepted: June 7, 1999?Published online December 15, 2000     

Some characterizations and properties of the “distance to ill-posedness” and the condition measure of a conic linear system

A conic linear system is a system of the form?P(d): find x that solves b - Ax∈C _Y , x∈C _X ,? where C _X and C _Y are closed convex cones, and the data for the system is d=(A,b). This system is“well-posed” to the extent that (small) changes in the data (A,b) do not alter the status of the system (the system remains solvable or not). Renegar defined the “distance to ill-posedness”, ρ(d), to be the smallest change in the data Δd=(ΔA,Δb) for which the system P(d+Δd) is “ill-posed”, i.e., d+Δd is in the intersection of the closure of feasible and infeasible instances d’=(A’,b’) of P(·). Renegar also defined the “condition measure” of the data instance d as C(d):=∥d∥/ρ(d), and showed that this measure is a natural extension of the familiar condition measure associated with systems of linear equations. This study presents two categories of results related to ρ(d), the distance to ill-posedness, and C(d), the condition measure of d. The first category of results involves the approximation of ρ(d) as the optimal value of certain mathematical programs. We present ten different mathematical programs each of whose optimal values provides an approximation of ρ(d) to within certain constants, depending on whether P(d) is feasible or not, and where the constants depend on properties of the cones and the norms used. The second category of results involves the existence of certain inscribed and intersecting balls involving the feasible region of P(d) or the feasible region of its alternative system, in the spirit of the ellipsoid algorithm. These results roughly state that the feasible region of P(d) (or its alternative system when P(d) is not feasible) will contain a ball of radius r that is itself no more than a distance R from the origin, where the ratio R/r satisfies R/r≤c ₁ C(d), and such that r≥ and R≤c ₃ C(d), where c ₁,c ₂,c ₃ are constants that depend only on properties of the cones and the norms used. Therefore the condition measure C(d) is a relevant tool in proving the existence of an inscribed ball in the feasible region of P(d) that is not too far from the origin and whose radius is not too small. Received November 2, 1995 / Revised version received June 26, 1998?Published online May 12, 1999