Nonconvex Piecewise-Quadratic Underestimation for Global Minimization

Motivated by the fact that important real-life problems, such as the protein docking problem, can be accurately modeled by minimizing a nonconvex piecewise-quadratic function, a nonconvex underestimator is constructed as the minimum of a finite number of strictly convex quadratic functions. The nonconvex underestimator is generated by minimizing a linear function on a reverse convex region and utilizes sample points from a given complex function to be minimized. The global solution of the piecewise-quadratic underestimator is known exactly and gives an approximation to the global minimum of the original function. Successive shrinking of the initial search region to which this procedure is applied leads to fairly accurate estimates, within 0.0060%, of the global minima of synthetic nonconvex functions for which the global minima are known. Furthermore, this process can approximate a nonconvex protein docking function global minimum within four-figure relative accuracy in six refinement steps. This is less than half the number of refinement steps required by previous models such as the convex kernel underestimator (Mangasarian et al., Computational Optimization and Applications, to appear) and produces higher accuracy here.     

Convex Kernel Underestimation of Functions with Multiple Local Minima

A function on Rⁿ with multiple local minima is approximated from below, via linear programming, by a linear combination of convex kernel functions using sample points from the given function. The resulting convex kernel underestimator is then minimized, using either a linear equation solver for a linear-quadratic kernel or by a Newton method for a Gaussian kernel, to obtain an approximation to a global minimum of the original function. Successive shrinking of the original search region to which this procedure is applied leads to fairly accurate estimates, within 0.0001% for a Gaussian kernel function, relative to global minima of synthetic nonconvex piecewise-quadratic functions for which the global minima are known exactly. Gaussian kernel underestimation improves by a factor of ten the relative error obtained using a piecewise-linear underestimator (O.L. Mangasarian, J.B. Rosen, and M.E. Thompson, Journal of Global Optimization, Volume 32, Number 1, Pages 1–9, 2005), while cutting computational time by an average factor of over 28.     

Iterative Convex Quadratic Approximation for Global Optimization in Protein Docking

An algorithm for finding an approximate global minimum of a funnel shaped function with many local minima is described. It is applied to compute the minimum energy docking position of a ligand with respect to a protein molecule. The method is based on the iterative use of a convex, general quadratic approximation that underestimates a set of local minima, where the error in the approximation is minimized in the L¹ norm. The quadratic approximation is used to generate a reduced domain, which is assumed to contain the global minimum of the funnel shaped function. Additional local minima are computed in this reduced domain, and an improved approximation is computed. This process is iterated until a convergence tolerance is satisfied. The algorithm has been applied to find the global minimum of the energy function generated by the Docking Mesh Evaluator program. Results for three different protein docking examples are presented. Each of these energy functions has thousands of local minima. Convergence of the algorithm to an approximate global minimum is shown for all three examples.     

Convex Underestimation of Twice Continuously Differentiable Functions by Piecewise Quadratic Perturbation: Spline αBB Underestimators

This paper describes the construction of convex underestimators for twice continuously differentiable functions over box domains through piecewise quadratic perturbation functions. A refinement of the classical α BB convex underestimator, the underestimators derived through this approach may be significantly tighter than the classical αBB underestimator. The convex underestimator is the difference of the nonconvex function f and a smooth, piecewise quadratic, perturbation function, q. The convexity of the underestimator is guaranteed through an analysis of the eigenvalues of the Hessian of f over all subdomains of a partition of the original box domain. Smoothness properties of the piecewise quadratic perturbation function are derived in a manner analogous to that of spline construction.     

A new global optimization method for univariate constrained twice-differentiable NLP problems

In this paper, a new global optimization method is proposed for an optimization problem with twice-differentiable objective and constraint functions of a single variable. The method employs a difference of convex underestimator and a convex cut function, where the former is a continuous piecewise concave quadratic function, and the latter is a convex quadratic function. The main objectives of this research are to determine a quadratic concave underestimator that does not need an iterative local optimizer to determine the lower bounding value of the objective function and to determine a convex cut function that effectively detects infeasible regions for nonconvex constraints. The proposed method is proven to have a finite ε-convergence to locate the global optimum point. The numerical experiments indicate that the proposed method competes with another covering method, the index branch-and-bound algorithm, which uses the Lipschitz constant.     

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.     

On jet-convex functions and their tensor products

In this article, we introduce necessary and sufficient conditions for the tensor product of two convex functions to be convex. For our analysis we introduce the notions of true convexity, jet-convexity, true jet-convexity as well as true log-convexity. The links between jet-convex and log-convex functions are elaborated. As an algebraic tool, we introduce the jet product of two symmetric matrices and study some of its properties. We illustrate our results by an application from global optimization, where a convex underestimator for the tensor product of two functions is constructed as the tensor product of convex underestimators of the single functions.     

Tight convex underestimators for {{\mathcal C}^2}-continuous problems: I. univariate functions

A novel method for the convex underestimation of univariate functions is presented in this paper. The method is based on a piecewise application of the well-known αBB underestimator, which produces an overall underestimator that is piecewise convex. Subsequently, two algorithms are used to identify the linear segments needed for the construction of its -continuous convex envelope, which is itself a valid convex underestimator of the original function. The resulting convex underestimators are very tight, and their tightness benefits from finer partitioning of the initial domain. It is theoretically proven that there is always some finite level of partitioning for which the method yields the convex envelope of the function of interest. The method was applied on a set of univariate test functions previously presented in the literature, and the results indicate that the method produces convex underestimators of high quality in terms of both lower bound and tightness over the whole domain under consideration.     

Trigonometric Convex Underestimator for the Base Functions in Fourier Space

A three-parameter (a, b, x_s) convex underestimator of the functional form (x) = -a sin[k(x-x_s)] + b for the function f(x) = sin(x+s), x [x_L, x_U], is presented. The proposed method is deterministic and guarantees the existence of at least one convex underestimator of this functional form. We show that, at small k, the method approaches an asymptotic solution. We show that the maximum separation distance of the underestimator from the minimum of the function grows linearly with the domain size. The method can be applied to trigonometric polynomial functions of arbitrary dimensionality and arbitrary degree. We illustrate the features of the new trigonometric underestimator with numerical examples.Support from the National Science Foundation and the National Institutes of Health Grant R01 GM52032 is gratefully acknowledged.     

Global Minimization via Piecewise-Linear Underestimation

Given a function on Rⁿ with many multiple local minima we approximate it from below, via concave minimization, with a piecewise-linear convex function by using sample points from the given function. The piecewise-linear function is then minimized using a single linear program to obtain an approximation to the global minimum of the original function. Successive shrinking of the original search region to which this procedure is applied leads to fairly accurate estimates, within 0.57%, of the global minima of synthetic nonconvex piecewise-quadratic functions for which the global minima are known exactly.     

A global optimization algorithm for linear fractional and bilinear programs

In this paper a deterministic method is proposed for the global optimization of mathematical programs that involve the sum of linear fractional and/or bilinear terms. Linear and nonlinear convex estimator functions are developed for the linear fractional and bilinear terms. Conditions under which these functions are nonredundant are established. It is shown that additional estimators can be obtained through projections of the feasible region that can also be incorporated in a convex nonlinear underestimator problem for predicting lower bounds for the global optimum. The proposed algorithm consists of a spatial branch and bound search for which several branching rules are discussed. Illustrative examples and computational results are presented to demonstrate the efficiency of the proposed algorithm.     

Computational Experience with a New Class of Convex Underestimators: Box-constrained NLP Problems

In Akrotirianakis and Floudas (2004) we presented the theoretical foundations of a new class of convex underestimators for C ² nonconvex functions. In this paper, we present computational experience with those underestimators incorporated within a Branch-and-Bound algorithm for box-conatrained problems. The algorithm can be used to solve global optimization problems that involve C ² functions. We discuss several ways of incorporating the convex underestimators within a Branch-and-Bound framework. The resulting Branch-and-Bound algorithm is then used to solve a number of difficult box-constrained global optimization problems. A hybrid algorithm is also introduced, which incorporates a stochastic algorithm, the Random-Linkage method, for the solution of the nonconvex underestimating subproblems, arising within a Branch-and-Bound framework. The resulting algorithm also solves efficiently the same set of test problems.     

Approximating Data in \mathfrak{R}^{n} by a Quadratic Underestimator with Specified Hessian Minimum and Maximum Eigenvalues

The problem of approximating m data points (x _i , y _i ) in , with a quadratic function q(x, p) with s parameters, m ≥ s, is considered. The parameter vector is to be determined so as to satisfy three conditions: (1) q(x, p) must underestimate all m data points, i.e. q(x _i , p) ≤ y _i , i=1,...,m. (2) The error of the approximation is to be minimized in the L1 norm. (3) The eigenvalues of H are to satisfy specified lower and upper bounds, where H is the Hessian of q(x, p) with respect to x. This is called the Quadratic Underestimator with Bounds on Eigenvalues (QUBE) problem. An algorithm for its solution (QUBE algorithm) is given and justified, and computational results presented. The QUBE algorithm has application to finding the global minimum of a basin (or funnel) shaped function with a large number of local minima. Such problems arise in computational biology where it is desired to find the global minimum of an energy surface, in order to predict native protein-ligand docking geometry (drug design) or protein structure. Computational results for a simulated docking energy surface, with n=15, are presented. It is shown that specifying a small condition number for H improves the ability of the underestimator to correctly predict the global minimum point.     

Minimal infeasible constraint sets in convex integer programs

In this paper we investigate certain aspects of infeasibility in convex integer programs, where the constraint functions are defined either as a composition of a convex increasing function with a convex integer valued function of n variables or the sum of similar functions. In particular we are concerned with the problem of an upper bound for the minimal cardinality of the irreducible infeasible subset of constraints defining the model. We prove that for the considered class of functions, every infeasible system of inequality constraints in the convex integer program contains an inconsistent subsystem of cardinality not greater than 2 ⁿ , this way generalizing the well known theorem of Scarf and Bell for linear systems. The latter result allows us to demonstrate that if the considered convex integer problem is bounded below, then there exists a subset of at most 2 ⁿ −1 constraints in the system, such that the minimum of the objective function subject to the inequalities in the reduced subsystem, equals to the minimum of the objective function over the entire system of constraints.     

Global minimum test problem construction

A method is presented for the construction of test problems for which the global minimum point is known.Given a bounded convex polyhedron inR ⁿ , and a selected vertex, a concave quadratic function is constructed which attains its global minimum at the selected vertex. In general, this function will also have many other local minima.This research was supported in part by NSF Grant MCS 81-01214.     

An alternative perspective on copositive and convex relaxations of nonconvex quadratic programs

We study convex relaxations of nonconvex quadratic programs. We identify a family of so-called feasibility preserving convex relaxations, which includes the well-known copositive and doubly nonnegative relaxations, with the property that the convex relaxation is feasible if and only if the nonconvex quadratic program is feasible. We observe that each convex relaxation in this family implicitly induces a convex underestimator of the objective function on the feasible region of the quadratic program. This alternative perspective on convex relaxations enables us to establish several useful properties of the corresponding convex underestimators. In particular, if the recession cone of the feasible region of the quadratic program does not contain any directions of negative curvature, we show that the convex underestimator arising from the copositive relaxation is precisely the convex envelope of the objective function of the quadratic program, strengthening Burer’s well-known result on the exactness of the copositive relaxation in the case of nonconvex quadratic programs. We also present an algorithmic recipe for constructing instances of quadratic programs with a finite optimal value but an unbounded relaxation for a rather large family of convex relaxations including the doubly nonnegative relaxation.

     

New Sufficiency for Global Optimality and Duality of Mathematical Programming Problems via Underestimators

We present new conditions for a Karush-Kuhn-Tucker point to be a global minimizer of a mathematical programming problem which may have many local minimizers that are not global. The new conditions make use of underestimators of the Lagrangian at the Karush-Kuhn-Tucker point. We establish that a Karush-Kuhn-Tucker point is a global minimizer if the Lagrangian admits an underestimator, which is convex or, more generally, has the property that every stationary point is a global minimizer. In particular, we obtain sufficient conditions by using the fact that the biconjugate function of the Lagrangian is a convex underestimator at a point whenever it coincides with the Lagrangian at that point. We present also sufficient conditions for weak and strong duality results in terms of underestimators. The authors are grateful to Professor Gue Myung Lee, Pukyong National University, Korea, and the referees for their comments and suggestions which have contributed to the final preparation of the paper. The work was partially supported by the Australian Research Council Discovery Project Grant.     

A Combined Global & Local Search (CGLS) Approach to Global Optimization

This paper presents a general approach that combines global search strategies with local search and attempts to find a global minimum of a real valued function of n variables. It assumes that derivative information is unreliable; consequently, it deals with derivative free algorithms, but derivative information can be easily incorporated. This paper presents a nonmonotone derivative free algorithm and shows numerically that it may converge to a better minimum starting from a local nonglobal minimum. This property is then incorporated into a random population to globalize the algorithm. Convergence to a zero order stationary point is established for nonsmooth convex functions, and convergence to a first order stationary point is established for strictly differentiable functions. Preliminary numerical results are encouraging. A Java implementation that can be run directly from the Web allows the interested reader to get a better insight of the performance of the algorithm on several standard functions. The general framework proposed here, allows the user to incorporate variants of well known global search strategies. Research done under the cooperation agreement between Universidade de Vigo and Universidad Simón Bolívar.     

Finding all solutions of nonlinearly constrained systems of equations

A new approach is proposed for finding all-feasible solutions for certain classes of nonlinearly constrained systems of equations. By introducing slack variables, the initial problem is transformed into a global optimization problem (P) whose multiple global minimum solutions with a zero objective value (if any) correspond to all solutions of the initial constrained system of equalities. All-globally optimal points of (P) are then localized within a set of arbitrarily small disjoint rectangles. This is based on a branch and bound type global optimization algorithm which attains finite-convergence to each of the multiple global minima of (P) through the successive refinement of a convex relaxation of the feasible region and the subsequent solution of a series of nonlinear convex optimization problems. Based on the form of the participating functions, a number of techniques for constructing this convex relaxation are proposed. By taking advantage of the properties of products of univariate functions, customized convex lower bounding functions are introduced for a large number of expressions that are or can be transformed into products of univariate functions. Alternative convex relaxation procedures involve either the difference of two convex functions employed in BB [23] or the exponential variable transformation based underestimators employed for generalized geometric programming problems [24]. The proposed approach is illustrated with several test problems. For some of these problems additional solutions are identified that existing methods failed to locate.     

A Note on the Alternating Direction Method of Multipliers

We consider the linearly constrained separable convex programming, whose objective function is separable into m individual convex functions without coupled variables. The alternating direction method of multipliers has been well studied in the literature for the special case m=2, while it remains open whether its convergence can be extended to the general case m≥3. This note shows the global convergence of this extension when the involved functions are further assumed to be strongly convex.