Global optimization for sum of generalized fractional functions

This paper considers the solution of generalized fractional programming (GFP) problem which contains various variants such as a sum or product of a finite number of ratios of linear functions, polynomial fractional programming, generalized geometric programming, etc. over a polytope. For such problems, we present an efficient unified method. In this method, by utilizing a transformation and a two-part linearization method, a sequence of linear programming relaxations of the initial nonconvex programming problem are derived which are embedded in a branch-and-bound algorithm. Numerical results are given to show the feasibility and effectiveness of the proposed algorithm.     

求一类非凸规划问题全局解的确定性算法

本文对一类非凸规划问题(NP)给出一确定性全局优化算法.这类问题包括:在非凸的可行域上极小化有限个带指数的线性函数乘积的和与差,广义线性多乘积规划,多项式规划等.通过利用等价问题和线性化技巧提出的算法收敛到问题(NP)的全局极小.     

An inexact projected gradient method with rounding and lifting by nonlinear programming for solving rank-one semidefinite relaxation of polynomial optimization

Mathematical Programming - We consider solving high-order and tight semidefinite programming (SDP) relaxations of nonconvex polynomial optimization problems (POPs) that often admit degenerate...     

Nonconvex piecewise linear knapsack problems

This paper considers the minimization version of a class of nonconvex knapsack problems with piecewise linear cost structure. The items to be included in the knapsack have a divisible quantity and a cost function. An item can be included partially in the given quantity range and the cost is a nonconvex piecewise linear function of quantity. Given a demand, the optimization problem is to choose an optimal quantity for each item such that the demand is satisfied and the total cost is minimized. This problem and its close variants are encountered in manufacturing planning, supply chain design, volume discount procurement auctions, and many other contemporary applications. Two separate mixed integer linear programming formulations of this problem are proposed and are compared with existing formulations. Motivated by different scenarios in which the problem is useful, the following algorithms are developed: (1) a fast polynomial time, near-optimal heuristic using convex envelopes; (2) exact pseudo-polynomial time dynamic programming algorithms; (3) a 2-approximation algorithm; and (4) a fully polynomial time approximation scheme. A comprehensive test suite is developed to generate representative problem instances with different characteristics. Extensive computational experiments show that the proposed formulations and algorithms are faster than the existing techniques.     

一类神经网络模型的稳定性

本文将一种求解凸规划问题的神经网络模型推广到求解一般的非凸非线性规划问题．理论分析表明;在适当的条件下,本文提出的求解非凸非线性规划问题的神经网络模型的平衡点是渐近稳定的,对应于非线性规划问题的局部最优解     

The sparsest solutions to <Emphasis Type="Italic">Z</Emphasis>-tensor complementarity problems

Finding the sparsest solutions to a tensor complementarity problem is generally NP-hard due to the nonconvexity and noncontinuity of the involved \(\ell _0\) norm. In this paper, a special type of tensor complementarity problems with Z-tensors has been considered. Under some mild conditions, we show that to pursuit the sparsest solutions is equivalent to solving polynomial programming with a linear objective function. The involved conditions guarantee the desired exact relaxation and also allow to achieve a global optimal solution to the relaxed nonconvex polynomial programming problem. Particularly, in comparison to existing exact relaxation conditions, such as RIP-type ones, our proposed conditions are easy to verify.     

A novel approach to Bilevel nonlinear programming

Recently developed methods of monotonic optimization have been applied successfully for studying a wide class of nonconvex optimization problems, that includes, among others, generalized polynomial programming, generalized multiplicative and fractional programming, discrete programming, optimization over the efficient set, complementarity problems. In the present paper the monotonic approach is extended to the General Bilevel Programming GBP Problem. It is shown that (GBP) can be transformed into a monotonic optimization problem which can then be solved by “polyblock” approximation or, more efficiently, by a branch-reduce-and-bound method using monotonicity cuts. The method is particularly suitable for Bilevel Convex Programming and Bilevel Linear Programming.      

Extended duality for nonlinear programming

Duality is an important notion for nonlinear programming (NLP). It provides a theoretical foundation for many optimization algorithms. Duality can be used to directly solve NLPs as well as to derive lower bounds of the solution quality which have wide use in other high-level search techniques such as branch and bound. However, the conventional duality theory has the fundamental limit that it leads to duality gaps for nonconvex problems, including discrete and mixed-integer problems where the feasible sets are generally nonconvex.     

The Euclidean Steiner tree problem in Rn: A mathematical programming formulation

A nonconvex mixed-integer programming formulation for the Euclidean Steiner Tree Problem (ESTP) in Rⁿ is presented. After obtaining separability between integer and continuous variables in the objective function, a Lagrange dual program is proposed. To solve this dual problem (and obtaining a lower bound for ESTP) we use subgradient techniques. In order to evaluate a subgradient at each iteration we have to solve three optimization problems, two in polynomial time, and one is a special convex nondifferentiable programming problem. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Branch-and-Bound Approach for Solving a Class of Generalized Semi-infinite Programming Problems

A nonconvex generalized semi-infinite programming problem is considered, involving parametric max-functions in both the objective and the constraints. For a fixed vector of parameters, the values of these parametric max-functions are given as optimal values of convex quadratic programming problems. Assuming that for each parameter the parametric quadratic problems satisfy the strong duality relation, conditions are described ensuring the uniform boundedness of the optimal sets of the dual problems w.r.t. the parameter. Finally a branch-and-bound approach is suggested transforming the problem of finding an approximate global minimum of the original nonconvex optimization problem into the solution of a finite number of convex problems.     

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.     

Global optimization algorithm for sum of generalized polynomial ratios problem

In this paper, a global optimization algorithm is proposed for solving sum of generalized polynomial ratios problem (P) which arises in various practical problems. Due to its intrinsic difficulty, less work has been devoted to globally solve the problem (P). For such problems, we present a branch and bound algorithm. In this method, by utilizing exponent transformation and new three-level linear relaxation method, a sequence of linear relaxation programming of the initial nonconvex programming problem (P) are derived which are embedded in a branch and bound algorithm. The proposed method need not introduce new variables and constraints and it is convergent to the global minimum of prime problem by means of the subsequent solutions of a series of linear programming problems. Several numerical examples in the literatures are tested to demonstrate that the proposed algorithm can systematically solve these examples to find the approximate ?-global optimum.     

Global optimization of general nonconvex problems with intermediate polynomial substructures

This work considers the global optimization of general nonconvex nonlinear and mixed-integer nonlinear programming problems with underlying polynomial substructures. We incorporate linear cutting planes inspired by reformulation-linearization techniques to produce tight subproblem formulations that exploit these underlying structures. These cutting plane strategies simultaneously convexify linear and nonlinear terms from multiple constraints and are highly effective at tightening standard linear programming relaxations generated by sequential factorable programming techniques. Because the number of available cutting planes increases exponentially with the number of variables, we implement cut filtering and selection strategies to prevent an exponential increase in relaxation size. We introduce algorithms for polynomial substructure detection, cutting plane identification, cut filtering, and cut selection and embed the proposed implementation in BARON at every node in the branch-and-bound tree. A computational study including randomly generated problems of varying size and complexity demonstrates that the exploitation of underlying polynomial substructures significantly reduces computational time, branch-and-bound tree size, and required memory.     

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.     

Exact Solutions of Some Nonconvex Quadratic Optimization Problems via SDP and SOCP Relaxations

We show that SDP (semidefinite programming) and SOCP (second order cone programming) relaxations provide exact optimal solutions for a class of nonconvex quadratic optimization problems. It is a generalization of the results by S. Zhang for a subclass of quadratic maximization problems that have nonnegative off-diagonal coefficient matrices of quadratic objective functions and diagonal coefficient matrices of quadratic constraint functions. A new SOCP relaxation is proposed for the class of nonconvex quadratic optimization problems by extracting valid quadratic inequalities for positive semidefinite cones. Its effectiveness to obtain optimal values is shown to be the same as the SDP relaxation theoretically. Numerical results are presented to demonstrate that the SOCP relaxation is much more efficient than the SDP relaxation.     

Implementations of special ordered sets in MP software

Special ordered sets (SOS) have been introduced as a practical device for efficiently handling special classes of nonconvex optimization problems. They are now implemented in most commercial codes for mathematical programming (MP software). The paper gives a survey of possible applications as multiple choice restrictions, conditional multiple choice restrictions, discrete variables, discontinuous variables and piecewise linear functions, global optimization of separable programming problems, alternative right-hand sides, overlapping special ordered sets and the solution of quadratic programming problems. Alternative problem formulations are discussed. Since special ordered sets are not defined uniquely modelling facilities depend on the definition of a special orderedset in a code. The paper demonstrates the superiority of SOS to the application of binary variables if they are treated judiciously.     

Properties and methods for finding the best rank-one approximation to higher-order tensors

The problem of finding the best rank-one approximation to higher-order tensors has extensive engineering and statistical applications. It is well-known that this problem is equivalent to a homogeneous polynomial optimization problem. In this paper, we study theoretical results and numerical methods of this problem, particularly focusing on the 4-th order symmetric tensor case. First, we reformulate the polynomial optimization problem to a matrix programming, and show the equivalence between these two problems. Then, we prove that there is no duality gap between the reformulation and its Lagrangian dual problem. Concerning the approaches to deal with the problem, we propose two relaxed models. The first one is a convex quadratic matrix optimization problem regularized by the nuclear norm, while the second one is a quadratic matrix programming regularized by a truncated nuclear norm, which is a D.C. function and therefore is nonconvex. To overcome the difficulty of solving this nonconvex problem, we approximate the nonconvex penalty by a convex term. We propose to use the proximal augmented Lagrangian method to solve these two relaxed models. In order to obtain a global solution, we propose an alternating least eigenvalue method after solving the relaxed models and prove its convergence. Numerical results presented in the last demonstrate, especially for nonpositive tensors, the effectiveness and efficiency of our proposed methods.     

Global optimization of signomial geometric programming problems

This paper presents a global optimization approach for solving signomial geometric programming problems. In most cases nonconvex optimization problems with signomial parts are difficult, NP-hard problems to solve for global optimality. But some transformation and convexification strategies can be used to convert the original signomial geometric programming problem into a series of standard geometric programming problems that can be solved to reach a global solution. The tractability and effectiveness of the proposed successive convexification framework is demonstrated by seven numerical experiments. Some considerations are also presented to investigate the convergence properties of the algorithm and to give a performance comparison of our proposed approach and the current methods in terms of both computational efficiency and solution quality.     

Generalized Lagrangian duality for nonconvex polynomial programs with polynomial multipliers

In this paper, under the existence of a certificate of nonnegativity of the objective function over the given constraint set, we present saddle-point global optimality conditions and a generalized Lagrangian duality theorem for (not necessarily convex) polynomial optimization problems, where the Lagrange multipliers are polynomials. We show that the nonnegativity certificate together with the archimedean condition guarantees that the values of the Lasserre hierarchy of semidefinite programming (SDP) relaxations of the primal polynomial problem converge asymptotically to the common primal–dual value. We then show that the known regularity conditions that guarantee finite convergence of the Lasserre hierarchy also ensure that the nonnegativity certificate holds and the values of the SDP relaxations converge finitely to the common primal–dual value. Finally, we provide classes of nonconvex polynomial optimization problems for which the Slater condition guarantees the required nonnegativity certificate and the common primal–dual value with constant multipliers and the dual problems can be reformulated as semidefinite programs. These classes include some separable polynomial programs and quadratic optimization problems with quadratic constraints that admit certain hidden convexity. We also give several numerical examples that illustrate our results.     

A global optimization using linear relaxation for generalized geometric programming

Many local optimal solution methods have been developed for solving generalized geometric programming (GGP). But up to now, less work has been devoted to solving global optimization of (GGP) problem due to the inherent difficulty. This paper considers the global minimum of (GGP) problems. By utilizing an exponential variable transformation and the inherent property of the exponential function and some other techniques the initial nonlinear and nonconvex (GGP) problem is reduced to a sequence of linear programming problems. The proposed algorithm is proven that it is convergent to the global minimum through the solutions of a series of linear programming problems. Test results indicate that the proposed algorithm is extremely robust and can be used successfully to solve the global minimum of (GGP) on a microcomputer.