A note on strong duality in convex semidefinite optimization: necessary and sufficient conditions

A strong duality which states that the optimal values of the primal convex problem and its Lagrangian dual problem are equal (i.e. zero duality gap) and the dual problem attains its maximum is a corner stone in convex optimization. In particular it plays a major role in the numerical solution as well as the application of convex semidefinite optimization. The strong duality requires a technical condition known as a constraint qualification (CQ). Several CQs which are sufficient for strong duality have been given in the literature. In this note we present new necessary and sufficient CQs for the strong duality in convex semidefinite optimization. These CQs are shown to be sharper forms of the strong conical hull intersection property (CHIP) of the intersecting sets of constraints which has played a critical role in other areas of convex optimization such as constrained approximation and error bounds. Research was partially supported by the Australian Research Council. The author is grateful to the referees for their helpful comments     

Linear programs with an additional reverse convex constraint

A constraintg(x)0 is said to be a reverse convex constraint if the functiong is continuous and strictly quasi-convex. The feasible regions for linear programs with an additional reverse convex constraint are generally non-convex and disconnected. It is shown that the convex hull of the feasible region is a convex polytope and, as a result, there is an optimal solution on an edge of the polytope defined by only the linear constraints. The only possible edges which can contain such an optimal solution are characterized in relation to the best feasible vertex of the polytope defined by only the linear constraints. This characterization then provides a finite algorithm for finding a globally optimal solution.Research supported by NSF Grant ENG76-12250 and ONR Contract N00014-78-C-0428.     

Aggregation via empirical risk minimization

Given a finite set F of estimators, the problem of aggregation is to construct a new estimator whose risk is as close as possible to the risk of the best estimator in F. It was conjectured that empirical minimization performed in the convex hull of F is an optimal aggregation method, but we show that this conjecture is false. Despite that, we prove that empirical minimization in the convex hull of a well chosen, empirically determined subset of F is an optimal aggregation method.     

Improved integer programming bounds using intersections of corner polyhedra

Consider the relaxation of an integer programming (IP) problem in which the feasible region is replaced by the intersection of the linear programming (LP) feasible region and the corner polyhedron for a particular LP basis. Recently a primal-dual ascent algorithm has been given for solving this relaxation. Given an optimal solution of this relaxation, we state criteria for selecting a new LP basis for which the associated relaxation is stronger. These criteria may be successively applied to obtain either an optimal IP solution or a lower bound on the cost of such a solution. Conditions are given for equality of the convex hull of feasible IP solutions and the intersection of all corner polyhedra.     

Computing Hereditary Convex Structures

Color red and blue the n vertices of a convex polytope \(\mathcal{P}\) in ?³. Can we compute the convex hull of each color class in o(nlog?n) time? What if we have more than two colors? What if the colors are random? Consider an arbitrary query halfspace and call the vertices of \(\mathcal{P}\) inside it blue: can the convex hull of the blue points be computed in time linear in their number? More generally, can we quickly compute the blue hull without looking at the whole polytope? This paper considers several instances of hereditary computation and provides new results for them. In particular, we resolve an eight-year old open problem by showing how to split a convex polytope in linear expected time.     

$$T_5$$-Configurations and non-rigid sets of matrices

In 2003 B. Kirchheim-D. Preiss constructed a Lipschitz map in the plane with 5 incompatible gradients, where incompatibility refers to the condition that no two of the five matrices are rank-one connected. The construction is via the method of convex integration and relies on a detailed understanding of the rank-one geometry resulting from a specific set of five matrices. The full computation of the rank-one convex hull for this specific set was later carried out in 2010 by W. Pompe in Calc. Var. PDE 37(3–4):461–473, (2010) by delicate geometric arguments. For more general sets of matrices a full computation of the rank-one convex hull is clearly out of reach. Therefore, in this short note we revisit the construction and propose a new, in some sense generic method for deciding whether convex integration for a given set of matrices can be carried out, which does not require the full computation of the rank-one convex hull.     

An FPTAS for minimizing the product of two non-negative linear cost functions

We consider a quadratic programming (QP) problem (Π) of the form min x ^T C x subject to Ax ≥ b, x ≥ 0 where \({C\in {\mathbb R}^{n \times n}_+, {\rm rank}(C)=1}\) and \({A\in {\mathbb R}^{m \times n}, b\in {\mathbb R}^m}\) . We present an fully polynomial time approximation scheme (FPTAS) for this problem by reformulating the QP (Π) as a parameterized LP and “rounding” the optimal solution. Furthermore, our algorithm returns an extreme point solution of the polytope. Therefore, our results apply directly to 0–1 problems for which the convex hull of feasible integer solutions is known such as spanning tree, matchings and sub-modular flows. They also apply to problems for which the convex hull of the dominant of the feasible integer solutions is known such as s, t-shortest paths and s, t-min-cuts. For the above discrete problems, the quadratic program Π models the problem of obtaining an integer solution that minimizes the product of two linear non-negative cost functions.     

An algorithm for convex constrained minimax optimization based on duality

We consider the problem of finding the minimum value of the upper hull ofn convex functionals on a Hilbert space, subject to convex constraints. The problem is reformulated as that of finding the minimum of the worst convex combination of these functionals, which eventually yields a saddle-point problem. We propose a new algorithm to solve this problem that simplifies the task of updating the dual variables. Simultaneously, the constraints can be dualized by introducing other dual multipliers. Convergence proofs are given and a concrete example shows the practical and computational advantages of the proposed algorithm and approach.This research has been supported by the Centre National de la Recherche Scientifique (CNRS-France) under Contract No. ATP-2340.     

Convex hull property and maximum principle for finite element minimisers of general convex functionals

The convex hull property is the natural generalization of maximum principles from scalar to vector valued functions. Maximum principles for finite element approximations are often crucial for the preservation of qualitative properties of the respective physical model. In this work we develop a convex hull property for $\mathbb{P }_1$ conforming finite elements on simplicial non-obtuse meshes. The proof does not resort to linear structures of partial differential equations but directly addresses properties of the minimiser of a convex energy functional. Therefore, the result holds for very general nonlinear partial differential equations including e.g. the $p$ -Laplacian and the mean curvature problem. In the case of scalar equations the introduce techniques can be used to prove standard discrete maximum principles for nonlinear problems. We conclude by proving a strong discrete convex hull property on strictly acute triangulations.     

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     

Starke konvexität helm standortproblem

It is shown, that the location-allocation problem posses a strongly convex objective function with a bound of convexity of m=(F/d ²)²/d. Here F and d are the area and the diameter of the convex hull formed by the "points" of the problem. In the case of the four-points-problem a better estimation is valid giving a sharp bound of convexity greater by the factor two. As means of proof the appendix contains two new, elementary geometric inequalities about the relations between the area and the and the diameter of a convex planar region with vertices lying in a certain angle-section.     

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.     

On solving discrete two-stage stochastic programs having mixed-integer first- and second-stage variables

In this paper, we propose a decomposition-based branch-and-bound (DBAB) algorithm for solving two-stage stochastic programs having mixed-integer first- and second-stage variables. A modified Benders' decomposition method is developed, where the Benders' subproblems define lower bounding second-stage value functions of the first-stage variables that are derived by constructing a certain partial convex hull representation of the two-stage solution space. This partial convex hull is sequentially generated using a convexification scheme such as the Reformulation-Linearization Technique (RLT) or lift-and-project process, which yields valid inequalities that are reusable in the subsequent subproblems by updating the values of the first-stage variables. A branch-and-bound algorithm is designed based on a hyperrectangular partitioning process, using the established property that any resulting lower bounding Benders' master problem defined over a hyperrectangle yields the same objective value as the original stochastic program over that region if the first-stage variable solution is an extreme point of the defining hyperrectangle or the second-stage solution satisfies the binary restrictions. We prove that this algorithm converges to a global optimal solution. Some numerical examples and computational results are presented to demonstrate the efficacy of this approach.     

Method of orienting curves for determining the convex hull of a finite set of points in the plane

In this article, we present an efficient algorithm to determine the convex hull of a finite planar set using the idea of the Method of Orienting Curves (introduced by Phu in Zur Lösung einer regulären Aufgabenklasse der optimalen Steuerung in Großen mittels Orientierungskurven, Optimization, 18 (1987), pp. 65–81, for solving optimal control problems with state constraints). The convex hull is determined by parts of orienting lines and a final line. Two advantages of this algorithm over some variations of Graham's convex hull algorithm are presented.     

Probabilities of hitting a convex hull

In this note, we consider the non-negative least-square method with a random matrix. This problem has connections with the probability that the origin is not in the convex hull of many random points. As related problems, suitable estimates are obtained as well on the probability that a small ball does not hit the convex hull.