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In this paper, we study the optimum correction of the absolute value equations through making minimal changes in the coefficient matrix and the right hand side vector and using spectral norm. This problem can be formulated as a non-differentiable, non-convex and unconstrained fractional quadratic programming problem. The regularized least squares is applied for stabilizing the solution of the fractional problem. The regularized problem is reduced to a unimodal single variable minimization problem and to solve it a bisection algorithm is proposed. The main difficulty of the algorithm is a complicated constraint optimization problem, for which two novel methods are suggested. We also present optimality conditions and bounds for the norm of the optimal solutions. Numerical experiments are given to demonstrate the effectiveness of suggested methods.

The purpose of this paper is to solve the equality-constrained minimization problem of polynomial functions. Let $${\mathbb {R}}$$ be the field of real numbers, and $${\mathbb {R}}[x_1,\ldots ,x_n]$$ the ring of polynomials over $${\mathbb {R}}$$ in variables $$x_1$$ , ..., $$x_n$$ . For an $$f\in {\mathbb {R}}[x_1,\ldots ,x_n]$$ and a finite subset H of $${\mathbb {R}}[x_1,\ldots ,x_n]$$ , denote by $${\mathscr {V}}(f:H)$$ the set $$\{f({\bar{\alpha }})\mid {\bar{\alpha }}\in {\mathbb {R}}^n, \hbox { and }h({\bar{\alpha }})=0,\,\forall h\in H\}$$ . In this paper, we provide some effective algorithms for computing the accurate value of the infimum $$\inf {\mathscr {V}}(f:H)$$ of $${\mathscr {V}}(f:H)$$ , deciding whether or not the constrained infimum $$\inf {\mathscr {V}}(f:H)$$ is attained when $$\inf {\mathscr {V}}(f:H)\ne \pm \infty $$ , and finding a point for the constrained minimum $$\min {\mathscr {V}}(f:H)$$ if $$\inf {\mathscr {V}}(f:H)$$ is attained. With the aid of the computer algebraic system Maple, our algorithms have been compiled into a general program to treat the equality-constrained minimization of polynomial functions with rational coefficients.

Partially sparse recovery problem is a special sparse recovery optimization problem with part of the support of the solution is known in advance. Such a problem is closely related to the problems on compressive sensing where the $$l_1$$ -norm of the whole solution vector is minimized. In this paper, we propose the partial range space property to obtain some uniqueness conditions on the solution of the partially sparse recovery problem, it is somewhat stronger than the range space property for general sparse recovery problem. And any pair of solution to partial $$l_0$$ -minimization problem satisfying the partial range space property is also the unique solution to partial $$l_1$$ -minimization problem under mild condition.

We study the single source localization problem which consists of minimizing the squared sum of the errors, also known as the maximum likelihood formulation of the problem. The resulting optimization model is not only nonconvex but is also nonsmooth. We first derive a novel equivalent reformulation as a smooth constrained nonconvex minimization problem. The resulting reformulation allows for deriving a delightfully simple algorithm that does not rely on smoothing or convex relaxations. The proposed algorithm is proven to generate bounded iterates which globally converge to critical points of the original objective function of the source localization problem. Numerical examples are presented to demonstrate the performance of our algorithm.

In this paper, the concepts of Pareto $H$-eigenvalue and Pareto $Z$-eigenvalue are introduced for studying constrained minimization problem and the necessary and sufficient conditions of such eigenvalues are given. It is proved that a symmetric tensor has at least one Pareto $H$-eigenvalue (Pareto $Z$-eigenvalue). Furthermore, the minimum Pareto $H$-eigenvalue (or Pareto $Z$-eigenvalue) of a symmetric tensor is exactly equal to the minimum value of constrained minimization problem of homogeneous polynomial deduced by such a tensor, which gives an alternative methods for solving the minimum value of constrained minimization problem. In particular, a symmetric tensor $\mathcal{A}$ is copositive if and only if every Pareto $H$-eigenvalue ($Z-$eigenvalue) of $\mathcal{A}$ is non-negative., Comment: 14 pages. arXiv admin note: text overlap with arXiv:1302.6084

In this paper, a proximal alternating direction method of multipliers is proposed for solving a minimization problem with Lipschitz nonconvex constraints. Such problems are raised in many engineering fields, such as the analytical global placement of very large scale integrated circuit design. The proposed method is essentially a new application of the classical proximal alternating direction method of multipliers. We prove that, under some suitable conditions, any subsequence of the sequence generated by the proposed method globally converges to a Karush---Kuhn---Tucker point of the problem. We also present a practical implementation of the method using a certain self-adaptive rule of the proximal parameters. The proposed method is used as a global placement method in a placer of very large scale integrated circuit design. Preliminary numerical results indicate that, compared with some state-of-the-art global placement methods, the proposed method is applicable and efficient.

In this paper, we introduce two iterative schemes (one implicit and one explicit) for finding a common element of the set of solutions of the generalized equilibrium problems and the set of all common fixed points of a nonexpansive semigroup in the framework of a real Hilbert space. We prove that both approaches converge strongly to a common element of such two sets. Such common element is the unique solution of a variational inequality, which is the optimality condition for a minimization problem. Furthermore, we utilize the main results to obtain two mean ergodic theorems for nonexpansive mappings in a Hilbert space. The results of this paper extend and improve the results of Li et al. (J Nonlinear Anal 70:3065---3071, 2009), Cianciaruso et al. (J Optim Theory Appl 146:491---509, 2010) and many others.

In this paper generalized type-I, generalized quasi type-I, generalized pseudo type-I and other related functions over cones are defined for a vector minimization problem. Sufficient optimality conditions are studied for this problem using Clarke's generalized gradients. A Mond-Weir type dual is formulated and weak and strong duality results are established.

In this article, we provide optimality conditions for global solutions to cubic minimization problems with box or binary constraints. Our main tool is an extension of the global subdifferential approach, developed by Jeyakumar et al. (J Glob Optim 36:471---481, 2007; Math Program A 110:521---541, 2007). We also derive optimality conditions that characterize global solutions completely in the case where the cubic objective function contains no cross terms. Examples are given to demonstrate that the optimality conditions can effectively be used for identifying global minimizers of certain cubic minimization problems with box or binary constraints.

In this article we discuss the conditions required to guarantee the non-emptiness and the boundedness of certain subsets of the set of Lagrange multipliers for an inequality and equality constrained vector minimization problem.

A kind of general convexification and concavification methods is proposed for solving some classes of global optimization problems with certain monotone properties. It is shown that these minimization problems can be transformed into equivalent concave minimization problem or reverse convex programming problem or canonical D.C. programming problem by using the proposed convexification and concavification schemes. The existing algorithms then can be used to find the global solutions of the transformed problems.

Jafari and Moslehi (J Glob Optim 54:389---404, 2012) state that certain single-machine scheduling problems with linear job deterioration are NP-hard. We show that this is not the case for the maximum lateness minimization problem and point out the issues in the analysis of Jafari and Moslehi (J Glob Optim 54:389---404, 2012).

A parallel stochastic algorithm is presented for solving the linearly constrained concave global minimization problem. The algorithm is a multistart method and makes use of a Bayesian stopping rule to identify the global minimum with high probability. Computational results are presented for more than 200 problems on a Cray X-MP EA/464 supercomputer.

A counterexample to an algorithm of Dien (1988) for solving a minimization problem with a quasiconcave objective function and both linear and a reverse-convex constraint shows that this algorithm needn't lead to a solution of the given problem.