On the convergence of two branch-and-bound algorithms for nonconvex programming problems

This note presents a new convergence property for each of two branch-and-bound algorithms for nonconvex programming problems (Falk-Soland algorithms and Horst algorithms). For each algorithm, it has been shown previously that, under certain conditions, whenever the algorithm generates an infinite sequence of points, at least one accumulation point of this sequence is a global minimum. We show here that, for each algorithm, in fact, under these conditions, every accumulation point of such a sequence is a global minimum.The author would like to thank Professor R. M. Soland for his helpful comments concerning this paper.     

A branch-and-bound algorithm for 0–1 parametric mixed integer programming

A branch-and-bound algorithm to solve 0–1 parametric mixed integer linear programming problems has been developed. The present algorithm is an extension of the branch-and-bound algorithm for parametric analysis on pure integer programming. The characteristic of the present method is that optimal solutions for all values of the parameter can be obtained.     

A parametric successive underestimation method for convex multiplicative programming problems

This paper addresses itself to the algorithm for minimizing the product of two nonnegative convex functions over a convex set. It is shown that the global minimum of this nonconvex problem can be obtained by solving a sequence of convex programming problems. The basic idea of this algorithm is to embed the original problem into a problem in a higher dimensional space and to apply a branch-and-bound algorithm using an underestimating function. Computational results indicate that our algorithm is efficient when the objective function is the product of a linear and a quadratic functions and the constraints are linear. An extension of our algorithm for minimizing the sum of a convex function and a product of two convex functions is also discussed.     

On duality bound methods for nonconvex global optimization

A counter-example is given to several recently published results on duality bound methods for nonconvex global optimization.     

On affine scaling algorithms for nonconvex quadratic programming

We investigate the use of interior algorithms, especially the affine-scaling algorithm, to solve nonconvex — indefinite or negative definite — quadratic programming (QP) problems. Although the nonconvex QP with a polytope constraint is a hard problem, we show that the problem with an ellipsoidal constraint is easy. When the hard QP is solved by successively solving the easy QP, the sequence of points monotonically converge to a feasible point satisfying both the first and the second order optimality conditions.Research supported in part by NSF Grant DDM-8922636 and the College Summer Grant, College of Business Administration, The University of Iowa.     

Generalized linear multiplicative and fractional programming

This paper addresses itself to the algorithm for minimizing the sum of a convex function and a product of two linear functions over a polytope. It is shown that this nonconvex minimization problem can be solved by solving a sequence of convex programming problems. The basic idea of this algorithm is to embed the original problem into a problem in higher dimension and apply a parametric programming (path following) approach. Also it is shown that the same idea can be applied to a generalized linear fractional programming problem whose objective function is the sum of a convex function and a linear fractional function.     

Optimality conditions for nonconvex semidefinite programming

This paper concerns nonlinear semidefinite programming problems for which no convexity assumptions can be made. We derive first- and second-order optimality conditions analogous to those for nonlinear programming. Using techniques similar to those used in nonlinear programming, we extend existing theory to cover situations where the constraint matrix is structurally sparse. The discussion covers the case when strict complementarity does not hold. The regularity conditions used are consistent with those of nonlinear programming in the sense that the conventional optimality conditions for nonlinear programming are obtained when the constraint matrix is diagonal. Received: May 15, 1998 / Accepted: April 12, 2000?Published online May 12, 2000     

A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations

Existing global optimization techniques for nonconvex quadratic programming (QP) branch by recursively partitioning the convex feasible set and thus generate an infinite number of branch-and-bound nodes. An open question of theoretical interest is how to develop a finite branch-and-bound algorithm for nonconvex QP. One idea, which guarantees a finite number of branching decisions, is to enforce the first-order Karush-Kuhn-Tucker (KKT) conditions through branching. In addition, such an approach naturally yields linear programming (LP) relaxations at each node. However, the LP relaxations are unbounded, a fact that precludes their use. In this paper, we propose and study semidefinite programming relaxations, which are bounded and hence suitable for use with finite KKT-branching. Computational results demonstrate the practical effectiveness of the method, with a particular highlight being that only a small number of nodes are required. This author was supported in part by NSF Grants CCR-0203426 and CCF-0545514.     

Efficient algorithms for solving rank two and rank three bilinear programming problems

Finitely convergent algorithms for solving rank two and three bilinear programming problems are proposed. A rank k bilinear programming problem is a nonconvex quadratic programming problem with the following structure: % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4baFfea0dXde9vqpa0lb9% cq0dXdb9IqFHe9FjuP0-iq0dXdbba9pe0lb9hs0dXda91qaq-xfr-x% fj-hmeGabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaacaWFTbGaa8% xAaiaa-5gacaWFPbGaa8xBaiaa-LgacaWF6bGaa8xzaiaa-bcacaWF% 7bacbiGaa43yamaaDaaaleaacaGFWaaabaGaa4hDaaaakiaa+Hhaca% GFRaGaa4hzamaaDaaaleaacaGFWaaabaGaa4hDaaaakiaa+LhacaGF% RaWaaabuaeaacaGFJbWaa0baaSqaaiaa+PgaaeaacaGF0baaaOGaam% iEaiabl+y6NjaadsgadaqhaaWcbaGaamOAaaqaaiaadshaaaGccaWG% 5bGaaiiFaaWcbaGaa8NAaiaa-1dacaWFXaaabeqdcqGHris5aOGaa4% hEaiabgIGiolaa+HfacaGFSaGaa4xEaiabgIGiolaa+LfacaWF9bGa% a8hlaaaa!5D2E!\[minimize \{ c_0^t x + d_0^t y + \sum\limits_{j = 1} {c_j^t xd_j^t y|} x \in X,y \in Y\} ,\]where X Rⁿ¹ and Y R ⁿ² are non-empty and bounded polytopes. We show that a variant of parametric simplex algorithm can solve large scale rank two bilinear programming problems efficiently. Also, we show that a cutting-cake algorithm, a more elaborate variant of parametric simplex algorithm can solve medium scale rank three problems.This research was supported in part by Grant-in-Aid for Scientific Research of the Ministry of Education, Science and Culture, Grant No. 63490010.     

A new reformulation-linearization technique for bilinear programming problems

This paper is concerned with the development of an algorithm for general bilinear programming problems. Such problems find numerous applications in economics and game theory, location theory, nonlinear multi-commodity network flows, dynamic assignment and production, and various risk management problems. The proposed approach develops a new Reformulation-Linearization Technique (RLT) for this problem, and imbeds it within a provably convergent branch-and-bound algorithm. The method first reformulates the problem by constructing a set of nonnegative variable factors using the problem constraints, and suitably multiplies combinations of these factors with the original problem constraints to generate additional valid nonlinear constraints. The resulting nonlinear program is subsequently linearized by defining a new set of variables, one for each nonlinear term. This RLT process yields a linear programming problem whose optimal value provides a tight lower bound on the optimal value to the bilinear programming problem. Various implementation schemes and constraint generation procedures are investigated for the purpose of further tightening the resulting linearization. The lower bound thus produced theoretically dominates, and practically is far tighter, than that obtained by using convex envelopes over hyper-rectangles. In fact, for some special cases, this process is shown to yield an exact linear programming representation. For the associated branch-and-bound algorithm, various admissible branching schemes are discussed, including one in which branching is performed by partitioning the intervals for only one set of variables x or y, whichever are fewer in number. Computational experience is provided to demonstrate the viability of the algorithm. For a large number of test problems from the literature, the initial bounding linear program itself solves the underlying bilinear programming problem.This paper was presented at the II. IIASA Workshop on Global Optimization, Sopron (Hungary), December 9–14, 1990.     

Bifurcation problems in nonlinear parametric programming

The nonlinear parametric programming problem is reformulated as a closed system of nonlinear equations so that numerical continuation and bifurcation techniques can be used to investigate the dependence of the optimal solution on the system parameters. This system, which is motivated by the Fritz John first-order necessary conditions, contains all Fritz John and all Karush-Kuhn-Tucker points as well as local minima and maxima, saddle points, feasible and nonfeasible critical points. Necessary and sufficient conditions for a singularity to occur in this system are characterized in terms of the loss of a complementarity condition, the linear dependence of the gradients of the active constraints, and the singularity of the Hessian of the Lagrangian on a tangent space. Any singularity can be placed in one of seven distinct classes depending upon which subset of these three conditions hold true at a solution. For problems with one parameter, we analyze simple and multiple bifurcation of critical points from a singularity arising from the loss of the complementarity condition, and then develop a set of conditions which guarantees the unique persistence of a minimum through this singularity. The research of this author was supported by National Science Foundation through NSF Grant DMS-85-10201 and by the Air Force Office of Scientific Research through instrument number AFOSR-ISSA-85-00079.     

Interior-point methods for nonconvex nonlinear programming: regularization and warmstarts

In this paper, we investigate the use of an exact primal-dual penalty approach within the framework of an interior-point method for nonconvex nonlinear programming. This approach provides regularization and relaxation, which can aid in solving ill-behaved problems and in warmstarting the algorithm. We present details of our implementation within the loqo algorithm and provide extensive numerical results on the CUTEr test set and on warmstarting in the context of quadratic, nonlinear, mixed integer nonlinear, and goal programming. Research of the first author is sponsored by ONR grant N00014-04-1-0145. Research of the second author is supported by NSF grant DMS-0107450.     

Zero duality gap for a class of nonconvex optimization problems

By an equivalent transformation using thepth power of the objective function and the constraint, a saddle point can be generated for a general class of nonconvex optimization problems. Zero duality gap is thus guaranteed when the primal-dual method is applied to the constructed equivalent form.The author very much appreciates the comments from Prof. Douglas J. White.     

Global optimization of a class of nonconvex quadratically constrained quadratic programming problems

In this paper we study a class of nonconvex quadratically constrained quadratic programming problems generalized from relaxations of quadratic assignment problems. We show that each problem is polynomially solved. Strong duality holds if a redundant constraint is introduced. As an application, a new lower bound is proposed for the quadratic assignment problem.     

Shadow prices in nonconvex mathematical programming

In this paper a definition is proposed for the concept of shadow prices in nonconvex programming. For a nonlinear program with equality and inequality constraints, existence of these prices and bounds for their possible values are obtained under the Mangasarian—Fromowitz regularity condition. Their exact values and some continuity properties are obtained under the more restrictive linear independence regularity condition. A definition of equilibrium prices is also proposed. Under convexity assumptions, all definitions and results coincide with those already known on this subject in convex programming.This research was supported by the Natural Sciences and Engineering Research Council of Canada under Grant A-9273.     

Bifurcations in parametric nonlinear programming

Bifurcation and continuation techniques are introduced as a class of methods for investigating the parametric nonlinear programming problem. Motivated by the Fritz John first-order necessary conditions, the parametric programming problem is first reformulated as a closed system of nonlinear equations which contains all Karush-Kuhn-Tucker and Fritz John points, both feasible and infeasible solutions, and relative minima, maxima, and saddle points. Since changes in the structure of the solution set and critical point type can occur only at singularities, necessary and sufficient conditions for the existence of a singularity are developed in terms of the loss of a complementarity condition, the linear dependence constraint qualification, and the singularity of the Hessian of the Lagrangian on a tangent space. After a brief introduction to elementary bifurcation theory, some simple singularities in this parametric problem are analyzed for both branching and persistence of local minima. Finally, a brief introduction to numerical continuation and bifurcation procedures is given to indicate how these facts can be used in a numerical investigation of the problem.This research was supported by the Air force Office of Scientific Research through grant number AFOSR-88-0059.     

A heuristic algorithm for nonlinear programming

In this paper, a heuristic algorithm for nonlinear programming is presented. The algorithm uses two search directions, and the Hessian of the Lagrangian function is approximated with the BFGS secant update. We show that the sequence of iterates convergeq-superlinearly if the sequence of approximating matrices satisfies a particular condition. Numerical results are presented.     

A bifurcation analysis of the nonlinear parametric programming problem

The structure of solutions to the nonlinear parametric programming problem with a one dimensional parameter is analyzed in terms of the bifurcation behavior of the curves of critical points and the persistence of minima along these curves. Changes in the structure of the solution occur at singularities of a nonlinear system of equations motivated by the Fritz John first-order necessary conditions. It has been shown that these singularities may be completely partitioned into seven distinct classes based upon the violation of one or more of the following: a complementarity condition, a constraint qualification, and the nonsingularity of the Hessian of the Lagrangian on a tangent space. To apply classical bifurcation techniques to these singularities, a further subdivision of each case is necessary. The structure of curves of critical points near singularities of lowest (zero) codimension within each case is analyzed, as well as the persistence of minima along curves emanating from these singularities. Bifurcation behavior is also investigated or discussed for many of the subcases giving rise to a codimension one singularity.This work was supported by the National Science Foundation through NSF Grants DMS-85-10201 and DMS-87-04679 and by the Air Force Office of Scientific Research through grant number AFOSR-88-0059.     

Interior-point methods for nonconvex nonlinear programming: Jamming and numerical testing

The paper considers an example of Wächter and Biegler which is shown to converge to a nonstationary point for the standard primal–dual interior-point method for nonlinear programming. The reason for this failure is analyzed and a heuristic resolution is discussed. The paper then characterizes the performance of LOQO, a line-search interior-point code, on a large test set of nonlinear programming problems. Specific types of problems which can cause LOQO to fail are identified.Research of the first and third authors supported by NSF grant DMS-9870317, ONR grant N00014-98-1-0036.Research of the second author supported by NSF grant DMS-9805495.     

Hidden convexity in some nonconvex quadratically constrained quadratic programming

We consider the problem of minimizing an indefinite quadratic objective function subject to twosided indefinite quadratic constraints. Under a suitable simultaneous diagonalization assumption (which trivially holds for trust region type problems), we prove that the original problem is equivalent to a convex minimization problem with simple linear constraints. We then consider a special problem of minimizing a concave quadratic function subject to finitely many convex quadratic constraints, which is also shown to be equivalent to a minimax convex problem. In both cases we derive the explicit nonlinear transformations which allow for recovering the optimal solution of the nonconvex problems via their equivalent convex counterparts. Special cases and applications are also discussed. We outline interior-point polynomial-time algorithms for the solution of the equivalent convex programs. This author's work was partially supported by GIF, the German-Israeli Foundation for Scientific Research and Development and by the Binational Science Foundation. This author's work was partially supported by National Science Foundation Grants DMS-9201297 and DMS-9401871.