A Global Optimization Method,QBB, for Twice-Differentiable Nonconvex Optimization Problem

A global optimization method, QBB, for twice-differentiable NLPs (Non-Linear Programming) is developed to operate within a branch-and-bound framework and require the construction of a relaxed convex problem on the basis of the quadratic lower bounding functions for the generic nonconvex structures. Within an exhaustive simplicial division of the constrained region, the rigorous quadratic underestimation function is constructed for the generic nonconvex function structure by virtue of the maximal eigenvalue analysis of the interval Hessian matrix. Each valid lower bound of the NLP problem with the division progress is computed by the convex programming of the relaxed optimization problem obtained by preserving the convex or linear terms, replacing the concave term with linear convex envelope, underestimating the special terms and the generic terms by using their customized tight convex lower bounding functions or the valid quadratic lower bounding functions, respectively. The standard convergence properties of the QBB algorithm for nonconvex global optimization problems are guaranteed. The preliminary computation studies are presented in order to evaluate the algorithmic efficiency of the proposed QBB approach.     

The Newton Bracketing Method for Convex Minimization

An iterative method for the minimization of convex functions f :ⁿ , called a Newton Bracketing (NB) method, is presented. The NB method proceeds by using Newton iterations to improve upper and lower bounds on the minimum value. The NB method is valid for n = 1, and in some cases for n > 1 (sufficient conditions given here). The NB method is applied to large scale Fermat–Weber location problems.     

A reformulation framework for global optimization

In this paper, we present a global optimization method for solving nonconvex mixed integer nonlinear programming (MINLP) problems. A convex overestimation of the feasible region is obtained by replacing the nonconvex constraint functions with convex underestimators. For signomial functions single-variable power and exponential transformations are used to obtain the convex underestimators. For more general nonconvex functions two versions of the so-called αBB-underestimator, valid for twice-differentiable functions, are integrated in the actual reformulation framework. However, in contrast to what is done in branch-and-bound type algorithms, no direct branching is performed in the actual algorithm. Instead a piecewise convex reformulation is used to convexify the entire problem in an extended variable-space, and the reformulated problem is then solved by a convex MINLP solver. As the piecewise linear approximations are made finer, the solution to the convexified and overestimated problem will form a converging sequence towards a global optimal solution. The result is an easily-implementable algorithm for solving a very general class of optimization problems.     

Global minimum potential energy conformations of small molecules

A global optimization algorithm is proposed for finding the global minimum potential energy conformations of small molecules. The minimization of the total potential energy is formulated on an independent set of internal coordinates involving only torsion (dihedral) angles. Analytical expressions for the Euclidean distances between non-bonded atoms, which are required for evaluating the individual pairwise potential terms, are obtained as functions of bond lengths, covalent bond angles, and torsion angles. A novel procedure for deriving convex lower bounding functions for the total potential energy function is also introduced. These underestimating functions satisfy a number of important theoretical properties. A global optimization algorithm is then proposed based on an efficient partitioning strategy which is guaranteed to attain -convergence to the global minimum potential energy configuration of a molecule through the solution of a series of nonlinear convex optimization problems. Moreover, lower and upper bounds on the total finite number of required iterations are also provided. Finally, this global optimization approach is illustrated with a number of example problems.     

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.     

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.     

Compositions of convex functions and fully linear models

Derivative-free optimization (DFO) is the mathematical study of the optimization algorithms that do not use derivatives. One branch of DFO focuses on model-based DFO methods, where an approximation of the objective function is used to guide the optimization algorithm. Proving convergence of such methods often applies an assumption that the approximations form fully linear models—an assumption that requires the true objective function to be smooth. However, some recent methods have loosened this assumption and instead worked with functions that are compositions of smooth functions with simple convex functions (the max-function or the \(\ell _1\) norm). In this paper, we examine the error bounds resulting from the composition of a convex lower semi-continuous function with a smooth vector-valued function when it is possible to provide fully linear models for each component of the vector-valued function. We derive error bounds for the resulting function values and subgradient vectors.     

Convex Envelopes of Monomials of Odd Degree

Convex envelopes of nonconvex functions are widely used to calculate lower bounds to solutions of nonlinear programming problems (NLP), particularly within the context of spatial Branch-and-Bound methods for global optimization. This paper proposes a nonlinear continuous and differentiable convex envelope for monomial terms of odd degree, x ^2k+1, where k N and the range of x includes zero. We prove that this envelope is the tightest possible. We also derive a linear relaxation from the proposed envelope, and compare both the nonlinear and linear formulations with relaxations obtained using other approaches.     

Convex Separable Minimization Subject to Bounded Variables

A minimization problem with convex and separable objective function subject to a separable convex inequality constraint and bounded variables is considered. A necessary and sufficient condition is proved for a feasible solution to be an optimal solution to this problem. Convex minimization problems subject to linear equality/linear inequality constraint, and bounds on the variables are also considered. A necessary and sufficient condition and a sufficient condition, respectively, are proved for a feasible solution to be an optimal solution to these two problems. Algorithms of polynomial complexity for solving the three problems are suggested and their convergence is proved. Some important forms of convex functions and computational results are given in the Appendix.     

D.C. programming approach for multicommodity network optimization problems with step increasing cost functions

We address a class of particularly hard-to-solve combinatorial optimization problems, namely that of multicommodity network optimization when the link cost functions are discontinuous step increasing. Unlike usual approaches consisting in the development of relaxations for such problems (in an equivalent form of a large scale mixed integer linear programming problem) in order to derive lower bounds, our d.c.(difference of convex functions) approach deals with the original continuous version and provides upper bounds. More precisely we approximate step increasing functions as closely as desired by differences of polyhedral convex functions and then apply DCA (difference of convex function algorithm) to the resulting approximate polyhedral d.c. programs. Preliminary computational experiments are presented on a series of test problems with structures similar to those encountered in telecommunication networks. They show that the d.c. approach and DCA provide feasible multicommodity flows x ^* such that the relative differences between upper bounds (computed by DCA) and simple lower bounds r:=(f(x^*)-LB)/{f(x^*)} lies in the range [4.2 %, 16.5 %] with an average of 11.5 %, where f is the cost function of the problem and LB is a lower bound obtained by solving the linearized program (that is built from the original problem by replacing step increasing cost functions with simple affine minorizations). It seems that for the first time so good upper bounds have been obtained.     

From error bounds to the complexity of first-order descent methods for convex functions

This paper shows that error bounds can be used as effective tools for deriving complexity results for first-order descent methods in convex minimization. In a first stage, this objective led us to revisit the interplay between error bounds and the Kurdyka-?ojasiewicz (KL) inequality. One can show the equivalence between the two concepts for convex functions having a moderately flat profile near the set of minimizers (as those of functions with Hölderian growth). A counterexample shows that the equivalence is no longer true for extremely flat functions. This fact reveals the relevance of an approach based on KL inequality. In a second stage, we show how KL inequalities can in turn be employed to compute new complexity bounds for a wealth of descent methods for convex problems. Our approach is completely original and makes use of a one-dimensional worst-case proximal sequence in the spirit of the famous majorant method of Kantorovich. Our result applies to a very simple abstract scheme that covers a wide class of descent methods. As a byproduct of our study, we also provide new results for the globalization of KL inequalities in the convex framework. Our main results inaugurate a simple method: derive an error bound, compute the desingularizing function whenever possible, identify essential constants in the descent method and finally compute the complexity using the one-dimensional worst case proximal sequence. Our method is illustrated through projection methods for feasibility problems, and through the famous iterative shrinkage thresholding algorithm (ISTA), for which we show that the complexity bound is of the form \(O(q^{k})\) where the constituents of the bound only depend on error bound constants obtained for an arbitrary least squares objective with \(\ell ^1\) regularization.     

An explicit characterization of the convex envelope of a bivariate bilinear function over special polytopes

In this paper, we constructively derive an explicit characterization of the convex envelope of a bilinear function over a special type of polytope in ². Our motivation stems from the use of such functions for deriving strengthened lower bounds within the context of a branch-and-bound algorithm for solving bilinear programming problems. For the case of polytopes with no edges having finite positive slopes, that is polytopes with downward sloping edges (which we call D-polytopes), we obtain a direct, explicit characterization of the convex envelope. This case subsumes the analysis of Al-Khayyal and Falk (1983) for constructing the convex envelope of a bilinear function over a rectangle in ². For non-D-polytopes, the analysis is more complex. We propose three strategies for this case based on (i) encasing the region in a D-polytope, (ii) employing a discretization technique, and (iii) providing an explicit characterization over a triangle along with a triangular decomposition approach. The analysis is illustrated using numerical examples.     

Primal-relaxed dual global optimization approach

A deterministic global optimization approach is proposed for nonconvex constrained nonlinear programming problems. Partitioning of the variables, along with the introduction of transformation variables, if necessary, converts the original problem into primal and relaxed dual subproblems that provide valid upper and lower bounds respectively on the global optimum. Theoretical properties are presented which allow for a rigorous solution of the relaxed dual problem. Proofs of -finite convergence and -global optimality are provided. The approach is shown to be particularly suited to (a) quadratic programming problems, (b) quadratically constrained problems, and (c) unconstrained and constrained optimization of polynomial and rational polynomial functions. The theoretical approach is illustrated through a few example problems. Finally, some further developments in the approach are briefly discussed.The authors gratefully acknowledge financial support from National Science Foundation Presidential Young Investigator Award CBT-88-57013. The authors are also grateful to Drs. F. A. Al-Khayyal, B. Jaumard, P. M. Pardalos, and H. D. Sherali for helpful comments on an earlier draft of this paper.     

Outer approximation algorithms for separable nonconvex mixed-integer nonlinear programs

A rigorous decomposition approach to solve separable mixed-integer nonlinear programs where the participating functions are nonconvex is presented. The proposed algorithms consist of solving an alternating sequence of Relaxed Master Problems (mixed-integer linear program) and two nonlinear programming problems (NLPs). A sequence of valid nondecreasing lower bounds and upper bounds is generated by the algorithms which converge in a finite number of iterations. A Primal Bounding Problem is introduced, which is a convex NLP solved at each iteration to derive valid outer approximations of the nonconvex functions in the continuous space. Two decomposition algorithms are presented in this work. On finite termination, the first yields the global solution to the original nonconvex MINLP and the second finds a rigorous bound to the global solution. Convergence and optimality properties, and refinement of the algorithms for efficient implementation are presented. Finally, numerical results are compared with currently available algorithms for example problems, illuminating the potential benefits of the proposed algorithm.     

Characterizing global optimality for DC optimization problems under convex inequality constraints

Characterizations of global optimality are given for general difference convex (DC) optimization problems involving convex inequality constraints. These results are obtained in terms of -subdifferentials of the objective and constraint functions and do not require any regularity condition. An extension of Farkas' lemma is obtained for inequality systems involving convex functions and is used to establish necessary and sufficient optimality conditions. As applications, optimality conditions are also given for weakly convex programming problems, convex maximization problems and for fractional programming problems.This paper was presented at the Optimization Miniconference held at the University of Ballarat, Victoria, Australia, on July 14, 1994.     

Higher-order total variation bounds for expectations of periodic functions and simple integer recourse approximations

We derive bounds on the expectation of a class of periodic functions using the total variations of higher-order derivatives of the underlying probability density function. These bounds are a strict improvement over those of Romeijnders et al. (Math Program 157:3–46, 2016b), and we use them to derive error bounds for convex approximations of simple integer recourse models. In fact, we obtain a hierarchy of error bounds that become tighter if the total variations of additional higher-order derivatives are taken into account. Moreover, each error bound decreases if these total variations become smaller. The improved bounds may be used to derive tighter error bounds for convex approximations of more general recourse models involving integer decision variables.     

Invariance of the order and type of a sequence of operators

In the paper, the invariance property of characteristics (the order and type) of an operator and of a sequence of operators with respect to a topological isomorphism is proved. These characteristics give precise upper and lower bounds for the expressions ‖A_n(x)‖_p and enable one to state and solve problems of operator theory in locally convex spaces in a general setting. Examples of such problems are given by the completeness problem for the set of values of a vector function in a locally convex space, the structure problem for a subspace invariant with respect to an operator A, the problem of applicability of an operator series to a locally convex space, the theory of holomorphic operator-valued functions, the theory of operator and differential-operator equations in nonnormed spaces, and so on. However, the immediate evaluation of characteristics of operators (and of sequences of operators) directly by definition is practically unrealizable in spaces with more complicated structure than that of countably normed spaces, due to the absence of an explicit form of seminorms or to their complicated structure. The approach that we use enables us to find characteristics of operators and sequences of operators using the passage to the dual space, by-passing the definition, and makes it possible to obtain bounds for the expressions ‖A_n(x)‖_p even if an explicit form of seminorms is unknown.     

Remarks on the generalized Newton method

We give some convergence results for the generalized Newton method for the computation of zeros of nondifferentiable functions which we proposed in an earlier work. Our results show that the generalized method can converge quadratically when used to compute the zeros of the sum of a differentiable function and the (multivalued) subgradient of a lower semicontinuous proper convex function. The method is therefore effective for variational inequalities and can be used to find the minimum of a function which is the sum of a twice-differentiable convex function and a lower semicontinuous proper convex function. A numerical example is given.     

A Rigorous Lower Bound for the Optimal Value of Convex Optimization Problems

In this paper, we consider the computation of a rigorous lower error bound for the optimal value of convex optimization problems. A discussion of large-scale problems, degenerate problems, and quadratic programming problems is included. It is allowed that parameters, whichdefine the convex constraints and the convex objective functions, may be uncertain and may vary between given lower and upper bounds. The error bound is verified for the family of convex optimization problems which correspond to these uncertainties. It can be used to perform a rigorous sensitivity analysis in convex programming, provided the width of the uncertainties is not too large. Branch and bound algorithms can be made reliable by using such rigorous lower bounds.     

New and efficient algorithms for transfer prices and inventory holding policies in two-enterprise supply chains

We consider a multi-period problem of fair transfer prices and inventory holding policies in two enterprise supply chains. This problem was formulated as a mixed integer non-linear program by Gjerdrum et al. (Eur J Oper Res 143:582–599, 2002). Existing global optimization methods to solve this problem are computationally expensive. We propose a continuous approach based on difference of convex functions (DC) programming and DC Algorithms (DCA) for solving this combinatorial optimization problem. The problem is first reformulated as a DC program via an exact penalty technique. Afterward, DCA, an efficient local approach in non-convex programming framework, is investigated to solve the resulting problem. For globally solving this problem, we investigate a combined DCA-Branch and Bound algorithm. DCA is applied to get lower bounds while upper bounds are computed from a relaxation problem. The numerical results on several test problems show that the proposed algorithms are efficient: DCA provides a good integer solution in a short CPU time although it works on a continuous domain, and the combined DCA-Branch and Bound finds an \(\epsilon \) -optimal solution for large-scale problems in a reasonable time.