The design centering problem as a D.C. programming problem

The following problem is studied: Given a compact setS inR ⁿ and a Minkowski functionalp(x), find the largest positive numberr for which there existsx S such that the set of ally R ⁿ satisfyingp(y–x) r is contained inS. It is shown that whenS is the intersection of a closed convex set and several complementary convex sets (sets whose complements are open convex) this design centering problem can be reformulated as the minimization of some d.c. function (difference of two convex functions) overR ⁿ . In the case where, moreover,p(x) = (x ^T Ax)^1/2, withA being a symmetric positive definite matrix, a solution method is developed which is based on the reduction of the problem to the global minimization of a concave function over a compact convex set.     

Computing a global optimal solution to a design centering problem

In this paper we present a method for solving a special three-dimensional design centering problem arising in diamond manufacturing: Find inside a given (not necessarily convex) polyhedral rough stone the largest diamond of prescribed shape and orientation. This problem can be formulated as the one of finding a global maximum of a difference of two convex functions over ³ and can be solved efficiently by using a global optimization algorithm provided that the objective function of the maximization problem can be easily evaluated. Here we prove that with the information available on the rough stone and on the reference diamond, evaluating the objective function at a pointx amounts to computing the distance, with respect to a Minkowski gauge, fromx to a finite number of planes. We propose a method for finding these planes and we report some numerical results.     

A note on functions whose local minima are global

In this note, we introduce a new class of generalized convex functions and show that a real functionf which is continuous on a compact convex subsetM of ⁿ and whose set of global minimizers onM is arcwise-connected has the property that every local minimum is global if, and only if,f belongs to that class of functions.     

Asymptotic behavior of the central path for a special class of degenerate SDP problems

This paper studies the asymptotic behavior of the central path (X(ν),S(ν),y(ν)) as ν↓0 for a class of degenerate semidefinite programming (SDP) problems, namely those that do not have strictly complementary primal-dual optimal solutions and whose “degenerate diagonal blocks” of the central path are assumed to satisfy We establish the convergence of the central path towards a primal-dual optimal solution, which is characterized as being the unique optimal solution of a certain log-barrier problem. A characterization of the class of SDP problems which satisfy our assumptions are also provided. It is shown that the re-parametrization t>0→(X(t⁴),S(t⁴),y(t⁴)) of the central path is analytic at t=0. The limiting behavior of the derivative of the central path is also investigated and it is shown that the order of convergence of the central path towards its limit point is Finally, we apply our results to the convex quadratically constrained convex programming (CQCCP) problem and characterize the class of CQCCP problems which can be formulated as SDPs satisfying the assumptions of this paper. In particular, we show that CQCCP problems with either a strictly convex objective function or at least one strictly convex constraint function lie in this class.This author was supported in part by CAPES and PRONEX-Otimização (FAPERJ/CNPq).This author was supported in part by FUNAPE/UFG, CAPES, PADCT-CNPq and PRONEX-Otimização (FAPERJ/CNPq).This author was supported in part by NSF Grants CCR-9902010, CCR-0203113 and INT-9910084 and ONR grant N00014-03-1-0401.Mathematics Subject Classification (1991): 90C20, 90C22, 90C25, 90C30, 90C33, 90C45, 90C51     

A maximization method for a class of quasiconcave programs

An implementable linearized method of centers is presented for solving a class of quasiconcave programs of the form (P): maximizef ⁰(x), subject tox B andf ⁱ (x)0, for everyi{1, ...,m}, whereB is a convex polyhedral subset ofR ⁿ (Euclideann-space). Each problem function is a continuous quasiconcave function fromR ⁿ intoR ¹. Also, it is assumed that the feasible region is bounded and there existsx B such thatf _i (x) for everyi {1, ...,m}. For a broad class of continuous quasiconcave problem functions, which may be nonsmooth, it is shown that the method produces a sequence of feasible points whose limit points are optimal for Problem (P). For many programs, no line searches are required. Additionally, the method is equipped with a constraint dropping devise.The author wishes to thank a referee for suggesting the use of generalized gradients and a second referee whose detailed informative comments have enhanced the paper.This work was done while the author was in the Department of Mathematical Sciences at the University of Delaware.     

A randomized scheme for speeding up algorithms for linear and convex programming problems with high constraints-to-variables ratio

We extend Clarkson's randomized algorithm for linear programming to a general scheme for solving convex optimization problems. The scheme can be used to speed up existing algorithms on problems which have many more constraints than variables. In particular, we give a randomized algorithm for solving convex quadratic and linear programs, which uses that scheme together with a variant of Karmarkar's interior point method. For problems withn constraints,d variables, and input lengthL, ifn = (d ²), the expected total number of major Karmarkar's iterations is O(d ²(logn)L), compared to the best known deterministic bound of O( L). We also present several other results which follow from the general scheme.     

A new algorithm for minimizing convex functions over convex sets

Let be a convex set for which there is an oracle with the following property. Given any pointz∈ℝ ⁿ the oracle returns a “Yes” ifz∈S; whereas ifz∉S then the oracle returns a “No” together with a hyperplane that separatesz fromS. The feasibility problem is the problem of finding a point inS; the convex optimization problem is the problem of minimizing a convex function overS. We present a new algorithm for the feasibility problem. The notion of a volumetric center of a polytope and a related ellipsoid of maximum volume inscribable in the polytope are central to the algorithm. Our algorithm has a significantly better global convergence rate and time complexity than the ellipsoid algorithm. The algorithm for the feasibility problem easily adapts to the convex optimization problem.     

Numerical decomposition of a convex function

Given then×p orthogonal matrixA and the convex functionf:R ⁿR, we find two orthogonal matricesP andQ such thatf is almost constant on the convex hull of ± the columns ofP, f is sufficiently nonconstant on the column space ofQ, and the column spaces ofP andQ provide an orthogonal direct sum decomposition of the column space ofA. This provides a numerically stable algorithm for calculating the cone of directions of constancy, at a pointx, of a convex function. Applications to convex programming are discussed.This work was supported by the National Science and Engineering Research Council of Canada (Grant No. A3388 and Summer Grant).     

Optimization of a convex program with a polynomial perturbation

We consider the problem of minimizing a convex function plus a polynomial p over a convex body K. We give an algorithm that outputs a solution x whose value is within range_K(p) of the optimum value, where range_K(p)=sup_xKp(x)−inf_xKp(x). When p depends only on a constant number of variables, the algorithm runs in time polynomial in 1/, the degree of p, the time to round K and the time to solve the convex program that results by setting p=0.     

A global optimization approach for solving the convex multiplicative programming problem

We consider a convex multiplicative programming problem of the form% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGG7bGaam% OzamaaBaaaleaacaaIXaaabeaakiaacIcacaWG4bGaaiykaiabgwSi% xlaadAgadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaamiEaiaacMcaca% GG6aGaamiEaiabgIGiolaadIfacaGG9baaaa!4A08!\[\{ f_1 (x) \cdot f_2 (x):x \in X\} \]where X is a compact convex set of ⁿ and f ₁, f ₂ are convex functions which have nonnegative values over X.Using two additional variables we transform this problem into a problem with a special structure in which the objective function depends only on two of the (n+2) variables. Following a decomposition concept in global optimization we then reduce this problem to a master problem of minimizing a quasi-concave function over a convex set in ² ₂. This master problem can be solved by an outer approximation method which requires performing a sequence of simplex tableau pivoting operations. The proposed algorithm is finite when the functions f _i, (i=1, 2) are affine-linear and X is a polytope and it is convergent for the general convex case.Partly supported by the Deutsche Forschungsgemeinschaft Project CONMIN.     

An algorithm for a singly constrained class of quadratic programs subject to upper and lower bounds

This paper gives an O(n) algorithm for a singly constrained convex quadratic program using binary search to solve the Kuhn-Tucker system. Computational results indicate that a randomized version of this algorithm runs in expected linear time and is suitable for practical applications. For the nonconvex case an-approximate algorithm is proposed which is based on convex and piecewise linear approximations of the objective function.     

On the classical logarithmic barrier function method for a class of smooth convex programming problems

In this paper, we describe a natural implementation of the classical logarithmic barrier function method for smooth convex programming. It is assumed that the objective and constraint functions fulfill the so-called relative Lipschitz condition, with Lipschitz constantM>0.In our method, we do line searches along the Newton direction with respect to the strictly convex logarithmic barrier function if we are far away from the central trajectory. If we are sufficiently close to this path, with respect to a certain metric, we reduce the barrier parameter. We prove that the number of iterations required by the algorithm to converge to an -optimal solution isO((1+M ²) log) orO((1+M ²)nlog), depending on the updating scheme for the lower bound.on leave from Eötvös University, Budapest, Hungary.     

Minimizing a quasi-concave function subject to a reverse convex constraint

Any constraintg(x)0 is called a reverse convex constraint ifg: R ⁿ R ¹ is a continuous convex function. This paper establishes a finite method for finding an optimal solution to a concave program with an additional reverse convex constraint. The method presented is a new approach to global optimization problems since it combines the idea of the branch and bound method with the idea of the cutting plane method.This paper is dedicated to Professor A. Pelczar     

A cutting plane algorithm for convex programming that uses analytic centers

Anoracle for a convex setS ⁿ accepts as input any pointz in ⁿ , and ifz S, then it returns yes, while ifz S, then it returns no along with a separating hyperplane. We give a new algorithm that finds a feasible point inS in cases where an oracle is available. Our algorithm uses the analytic center of a polytope as test point, and successively modifies the polytope with the separating hyperplanes returned by the oracle. The key to establishing convergence is that hyperplanes judged to be unimportant are pruned from the polytope. If a ball of radius 2^–L is contained inS, andS is contained in a cube of side 2 ^L+1, then we can show our algorithm converges after O(nL ²) iterations and performs a total of O(n ⁴ L ³+TnL ²) arithmetic operations, whereT is the number of arithmetic operations required for a call to the oracle. The bound is independent of the number of hyperplanes generated in the algorithm. An important application in which an oracle is available is minimizing a convex function overS. Supported by the National Science Foundation under Grant CCR-9057481PYI.Supported by the National Science Foundation under Grants CCR-9057481 and CCR-9007195.     

Convex Quadratic Approximation

For some applications it is desired to approximate a set of m data points in ⁿ with a convex quadratic function. Furthermore, it is required that the convex quadratic approximation underestimate all m of the data points. It is shown here how to formulate and solve this problem using a convex quadratic function with s = (n + 1)(n + 2)/2 parameters, s m, so as to minimize the approximation error in the L ¹ norm. The approximating function is q(p,x), where p ^s is the vector of parameters, and x n. The Hessian of q(p,x) with respect to x (for fixed p) is positive semi-definite, and its Hessian with respect to p (for fixed x) is shown to be positive semi-definite and of rank n. An algorithm is described for computing an optimal p* for any specified set of m data points, and computational results (for n = 4,6,10,15) are presented showing that the optimal q(p*,x) can be obtained efficiently. It is shown that the approximation will usually interpolate s of the m data points.     

Mathematical programs with a two-dimensional reverse convex constraint

We consider the problem min {f(x): x G, T(x) int D}, where f is a lower semicontinuous function, G a compact, nonempty set in ⁿ, D a closed convex set in ² with nonempty interior and T a continuous mapping from ⁿ to ². The constraint T(x) int D is a reverse convex constraint, so the feasible domain may be disconnected even when f, T are affine and G is a polytope. We show that this problem can be reduced to a quasiconcave minimization problem over a compact convex set in ² and hence can be solved effectively provided f, T are convex and G is convex or discrete. In particular we discuss a reverse convex constraint of the form c, x · d, x1. We also compare the approach in this paper with the parametric approach.     

Convexity of Nonlinear Image of a Small Ball with Applications to Optimization

Let f: XY be a nonlinear differentiable map, X,Y are Hilbert spaces, B(a,r) is a ball in X with a center a and radius r. Suppose f (x) is Lipschitz in B(a,r) with Lipschitz constant L and f (a) is a surjection: f (a)X=Y; this implies the existence of >0 such that f (a)^* yy, yY. Then, if r,/(2L), the image F=f(B(a,)) of the ball B(a,) is convex. This result has numerous applications in optimization and control. First, duality theory holds for nonconvex mathematical programming problems with extra constraint x–a. Special effective algorithms for such optimization problems can be constructed as well. Second, the reachability set for small power control is convex. This leads to various results in optimal control.     

A projection and contraction method for a class of linear complementarity problems and its application in convex quadratic programming

In this paper we propose a new iterative method for solving a class of linear complementarity problems:u 0,Mu + q 0, u^T(Mu + q)=0, where M is a givenl ×l positive semidefinite matrix (not necessarily symmetric) andq is a givenl-vector. The method makes two matrix-vector multiplications and a trivial projection onto the nonnegative orthant at each iteration, and the Euclidean distance of the iterates to the solution set monotonously converges to zero. The main advantages of the method presented are its simplicity, robustness, and ability to handle large problems with any start point. It is pointed out that the method may be used to solve general convex quadratic programming problems. Preliminary numerical experiments indicate that this method may be very efficient for large sparse problems.On leave from the Department of Mathematics, University of Nanjing, Nanjing, People's Republic of China.     

Polyhedral results on single node variable upper-bound flow models with allowed configurations

In this paper we investigate the convex hull of single node variable upper-bound flow models with allowed configurations. Such a model is defined by a set , where ρ is one of , = or , and Z{0,1}ⁿ consists of the allowed configurations. We consider the case when Z consists of affinely independent vectors. Under this assumption, a characterization of the non-trivial facets of the convex hull of X_ρ(Z) for each relation ρ is provided, along with polynomial time separation algorithms. Applications in scheduling and network design are also discussed.     

Convex Minimization on a Grid and Applications

This article discusses a discrete version of the convex minimization problem with applications to the efficient computation of proximity measures for pairs of convex polyhedra. Given ad-variate convex function and an isothetic grid of sizeO(n^d) in ^d, which is supposed to be finite but not necessarily regular, we want to find the grid cell containing the minimum point. With this aim, we identify a class of elementary subproblems, each resulting in the determination of a half-space in ^d, and show that the minimization problem can be solved by computingO(log n) half-spaces in the worst case foralmost uniformgrids of fixed dimensiondandO(log n) half-planes in the average for arbitrary planar grids. A major point is the potential of the approach to uniformly solve distance related problems for different configurations of a pair of convex bodies. In this respect, the case of a bivariate function is of particular interest and leads to a fast algorithm for detecting collisions between two convex polyhedra in three dimensions. The collision algorithm runs inO(log² n) average time for polyhedra withO(n) vertices whose boundaries are suitably represented; more specifically, the 1-skeletons can be embedded into layered Directed Acyclic Graphs which require justO(n) storage. The article ends with a brief discussion of a few experimental results.