Analytic efficient solution set for multi-criteria quadratic programs

We present active set methods to evaluate the exact analytic efficient solution set for multi-criteria convex quadratic programming problems (MCQP) subject to linear constraints. The idea is based on the observations that a strictly convex programming problem admits a unique solution, and that the efficient solution set for a multi-criteria strictly convex quadratic programming problem with linear equality constraints can be parameterized. The case of bi-criteria quadratic programming (BCQP) is first discussed since many of the underlying ideas can be explained much more clearly in the case of two objectives. In particular we note that the efficient solution set of a BCQP problem is a curve on the surface of a polytope. The extension to problems with more than two objectives is straightforward albeit some slightly more complicated notation. Two numerical examples are given to illustrate the proposed methods.     

On the convergence of inexact augmented Lagrangian methods for problems with convex constraints

We consider the augmented Lagrangian method (ALM) for constrained optimization problems in the presence of convex inequality and convex abstract constraints. We focus on the case where the Lagrangian sub-problems are solved up to approximate stationary points, with increasing accuracy. We analyze two different criteria of approximate stationarity for the sub-problems and we prove the global convergence to stationary points of ALM in both cases.     

METHOD OF CENTERS ALGORITHM FOR MULTI-OBJECTIVE PROGRAMMING PROBLEMS

In this paper, we consider a method of centers for solving multi-objective programming problems, where the objective functions involved are concave functions and the set of feasible points is convex. The algorithm is defined so that the sub-problems that must be solved during its execution may be solved by finite-step procedures. Conditions are given under which the algorithm generates sequences of feasible points and constraint multiplier vectors that have accumulation points satisfying the KKT conditions. Finally, we establish convergence of the proposed method of centers algorithm for solving multiobjective programming problems.     

Bi-criteria optimization problems for decision rules

We consider bi-criteria optimization problems for decision rules and rule systems relative to length and coverage. We study decision tables with many-valued decisions in which each row is associated with a set of decisions as well as single-valued decisions where each row has a single decision. Short rules are more understandable; rules covering more rows are more general. Both of these problems—minimization of length and maximization of coverage of rules are NP-hard. We create dynamic programming algorithms which can find the minimum length and the maximum coverage of rules, and can construct the set of Pareto optimal points for the corresponding bi-criteria optimization problem. This approach is applicable for medium-sized decision tables. However, the considered approach allows us to evaluate the quality of various heuristics for decision rule construction which are applicable for relatively big datasets. We can evaluate these heuristics from the point of view of (i) single-criterion—we can compare the length or coverage of rules constructed by heuristics; and (ii) bi-criteria—we can measure the distance of a point (length, coverage) corresponding to a heuristic from the set of Pareto optimal points. The presented results show that the best heuristics from the point of view of bi-criteria optimization are not always the best ones from the point of view of single-criterion optimization.     

Motzkin decomposition of closed convex sets

Theodore Motzkin proved, in 1936, that any polyhedral convex set can be expressed as the (Minkowski) sum of a polytope and a polyhedral convex cone. This paper provides five characterizations of the larger class of closed convex sets in finite dimensional Euclidean spaces which are the sum of a compact convex set with a closed convex cone. These characterizations involve different types of representations of closed convex sets as the support functions, dual cones and linear systems whose relationships are also analyzed in the paper. The obtaining of information about a given closed convex set F and the parametric linear optimization problem with feasible set F from each of its different representations, including the Motzkin decomposition, is also discussed.     

Convex solution of a permutation problem

In this paper, we show that a problem of finding a permuted version of k vectors from R^N such that they belong to a prescribed rank r subset, can be solved by convex optimization. We prove that under certain generic conditions, the wanted permutation matrix is unique in the convex set of doubly-stochastic matrices. In particular, this implies a solution of the classical correspondence problem of finding a permutation that transforms one collection of points in R^k into the another one. Solutions to these problems have a wide set of applications in Engineering and Computer Science.     

The Lipschitzianity of convex vector and set-valued functions

It is well known that every scalar convex function is locally Lipschitz on the interior of its domain in finite dimensional spaces. The aim of this paper is to extend this result for both vector functions and set-valued mappings acting between infinite dimensional spaces with an order generated by a proper convex cone C. Under the additional assumption that the ordering cone C is normal, we prove that a locally C-bounded C-convex vector function is Lipschitz on the interior of its domain by two different ways. Moreover, we derive necessary conditions for Pareto minimal points of vector-valued optimization problems where the objective function is C-convex and C-bounded. Corresponding results are derived for set-valued optimization problems.     

A solution to Hammer's X-ray reconstruction problem

We propose algorithms for reconstructing a planar convex body K from possibly noisy measurements of either its parallel X-rays taken in a fixed finite set of directions or its point X-rays taken at a fixed finite set of points, in known situations that guarantee a unique solution when the data is exact. The algorithms construct a convex polygon Pk whose X-rays approximate (in the least squares sense) k equally spaced noisy X-ray measurements in each of the directions or at each of the points.It is shown that these procedures are strongly consistent, meaning that, almost surely, Pk tends to K in the Hausdorff metric as k→∞. This solves, for the first time in the strongest sense, Hammer's X-ray problem published in 1963.     

Bundle methods for sum-functions with “easy” components: applications to multicommodity network design

We propose a version of the bundle scheme for convex nondifferentiable optimization suitable for the case of a sum-function where some of the components are “easy”, that is, they are Lagrangian functions of explicitly known compact convex programs. This corresponds to a stabilized partial Dantzig–Wolfe decomposition, where suitably modified representations of the “easy” convex subproblems are inserted in the master problem as an alternative to iteratively inner-approximating them by extreme points, thus providing the algorithm with exact information about a part of the dual objective function. The resulting master problems are potentially larger and less well-structured than the standard ones, ruling out the available specialized techniques and requiring the use of general-purpose solvers for their solution; this strongly favors piecewise-linear stabilizing terms, as opposed to the more usual quadratic ones, which in turn may have an adverse effect on the convergence speed of the algorithm, so that the overall performance may depend on appropriate tuning of all these aspects. Yet, very good computational results are obtained in at least one relevant application: the computation of tight lower bounds for Fixed-Charge Multicommodity Min-Cost Flow problems.     

Enhanced Efficiency in Multi-objective Optimization

For a given multi-objective optimization problem, we introduce and study the notion of α-proper efficiency. We give two characterizations of such proper efficiency: one is in terms of exact penalization and the other is in terms of stability of associated parametric problems. Applying the aforementioned characterizations and recent results on global error bounds for inequality systems, we obtain verifiable conditions for α-proper efficiency. For a large class of polynomial multi-objective optimization problems, we show that any efficient solution is α-properly efficient under some mild conditions. For a convex quadratically constrained multi-objective optimization problem with convex quadratic objective functions, we show that any efficient solution is α-properly efficient with a known estimate on α whenever its constraint set is bounded. Finally, we illustrate our achieved results with examples, and give an example to show that such an enhanced efficiency property may not hold for multi-objective optimization problems involving C ^∞-functions as objective functions.     

Acceleration of conjugate gradient algorithms for unconstrained optimization

Conjugate gradient methods are important for large-scale unconstrained optimization. This paper proposes an acceleration of these methods using a modification of steplength. The idea is to modify in a multiplicative manner the steplength α_k, computed by Wolfe line search conditions, by means of a positive parameter η_k, in such a way to improve the behavior of the classical conjugate gradient algorithms. It is shown that for uniformly convex functions the convergence of the accelerated algorithm is still linear, but the reduction in function values is significantly improved. Numerical comparisons with some conjugate gradient algorithms using a set of 750 unconstrained optimization problems, some of them from the CUTE library, show that the accelerated computational scheme outperform the corresponding conjugate gradient algorithms.     

New error bounds and their applications to convergence analysis of iterative algorithms

We present two new error bounds for optimization problems over a convex set whose objective function f is either semianalytic or γ-strictly convex, with γ≥1. We then apply these error bounds to analyze the rate of convergence of a wide class of iterative descent algorithms for the aforementioned optimization problem. Our analysis shows that the function sequence {f(x ^k )} converges at least at the sublinear rate of k ^-ε for some positive constant ε, where k is the iteration index. Moreover, the distances from the iterate sequence {x ^k } to the set of stationary points of the optimization problem converge to zero at least sublinearly. Received: October 5, 1999 / Accepted: January 1, 2000?Published online July 20, 2000     

Computing Proximal Points of Convex Functions with Inexact Subgradients

Locating proximal points is a component of numerous minimization algorithms. This work focuses on developing a method to find the proximal point of a convex function at a point, given an inexact oracle. Our method assumes that exact function values are at hand, but exact subgradients are either not available or not useful. We use approximate subgradients to build a model of the objective function, and prove that the method converges to the true prox-point within acceptable tolerance. The subgradient g _k used at each step k is such that the distance from g _k to the true subdifferential of the objective function at the current iteration point is bounded by some fixed ε > 0. The algorithm includes a novel tilt-correct step applied to the approximate subgradient.     

An approximation algorithm for convex multi-objective programming problems

In multi-objective convex optimization it is necessary to compute an infinite set of nondominated points. We propose a method for approximating the nondominated set of a multi-objective nonlinear programming problem, where the objective functions and the feasible set are convex. This method is an extension of Benson’s outer approximation algorithm for multi-objective linear programming problems. We prove that this method provides a set of weakly ε-nondominated points. For the case that the objectives and constraints are differentiable, we describe an efficient way to carry out the main step of the algorithm, the construction of a hyperplane separating an exterior point from the feasible set in objective space. We provide examples that show that this cannot always be done in the same way in the case of non-differentiable objectives or constraints.     

Newton-like methods for solving vector optimization problems

In the context of Euclidean spaces, we present an extension of the Newton-like method for solving vector optimization problems, with respect to the partial orders induced by a pointed, closed and convex cone with a nonempty interior. We study both exact and inexact versions of the Newton-like method. Under reasonable hypotheses, we prove stationarity of accumulation points of the sequences produced by Newton-like methods. Moreover, assuming strict cone-convexity of the objective map to the vector optimization problem, we establish convergence of the sequences to an efficient point whenever the initial point is in a compact level set.     

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.     

A bi-criteria two-machine flowshop scheduling problem with a learning effect

This paper addresses a bi-criteria two-machine flowshop scheduling problem when the learning effect is present. The objective is to find a sequence that minimizes a weighted sum of the total completion time and the maximum tardiness. In this article, a branch-and-bound method, incorporating several dominance properties and a lower bound, is presented to search for the exact solution for small job-size problems. In addition, two heuristic algorithms are proposed to overcome the inefficiency of the branch-and-bound algorithm for large job-size problems. Finally, computational results for this problem are provided to evaluate the performance of the proposed algorithms.     

Design of planar articulated mechanisms using branch and bound

This paper considers an optimization model and a solution method for the design of two-dimensional mechanical mechanisms. The mechanism design problem is modeled as a nonconvex mixed integer program which allows the optimal topology and geometry of the mechanism to be determined simultaneously. The underlying mechanical analysis model is based on a truss representation allowing for large displacements. For mechanisms undergoing large displacements elastic stability is of major concern. We derive conditions, modeled by nonlinear matrix inequalities, which guarantee that a stable equilibrium is found and that buckling is prevented. The feasible set of the design problem is described by nonlinear differentiable and non-differentiable constraints as well as nonlinear matrix inequalities.To solve the mechanism design problem a branch and bound method based on convex relaxations is developed. To guarantee convergence of the method, two different types of convex relaxations are derived. The relaxations are strengthened by adding valid inequalities to the feasible set and by solving bound contraction sub-problems. Encouraging computational results indicate that the branch and bound method can reliably solve mechanism design problems of realistic size to global optimality.     

Tikhonov-type regularization method for efficient solutions in vector optimization

The paper is devoted to developing the Tikhonov-type regularization algorithm of finding efficient solutions to the vector optimization problem for a mapping between finite dimensional Hilbert spaces with respect to the partial order induced by a pointed closed convex cone. We prove that under some suitable conditions either the sequence generated by our method converges to an efficient solution or all of its cluster points belong to the set of all efficient solutions of this problem.     

An Algorithm for Solving the Minimum-Norm Point Problem over the Intersection of a Polytope and an Affine Set

In this paper, we propose an efficient algorithm for finding the minimum-norm point in the intersection of a polytope and an affine set in an n-dimensional Euclidean space, where the polytope is expressed as the convex hull of finitely many points and the affine set is expressed as the intersection of k hyperplanes, k1. Our algorithm solves the problem by using directly the original points and the hyperplanes, rather than treating the problem as a special case of the general quadratic programming problem. One of the advantages of our approach is that our algorithm works as well for a class of problems with a large number (possibly exponential or factorial in n) of given points if every linear optimization problem over the convex hull of the given points is solved efficiently. The problem considered here is highly degenerate, and we take care of the degeneracy by solving a subproblem that is a conical version of the minimum-norm point problem, where points are replaced by rays. When the number k of hyperplanes expressing the affine set is equal to one, we can easily avoid degeneracy, but this is not the case for k2. We give a subprocedure for treating the degenerate case. The subprocedure is interesting in its own right. We also show the practical efficiency of our algorithm by computational experiments.