On the connections between semidefinite optimization and vector optimization

We use asymptotic analysis to develop finer estimates for the efficient, weak efficient and proper efficient solution sets (and for their asymptotic cones) to convex/quasiconvex vector optimization problems. We also provide a new representation for the efficient solution set without any convexity assumption, and the estimates involve the minima of the linear scalarization of the original vector problem. Some new necessary conditions for a point to be efficient or weak efficient solution for general convex vector optimization problems, as well as for the nonconvex quadratic multiobjective optimization problem, are established.

The aim of this article is to develop an approximate (ε)-conjugate duality theory for a general vector optimization problem with set-valued maps on the basis of ε-weak efficiency. We first introduce the concepts of ε-conjugate maps and ε-weak subgradients for set-valued maps and establish several properties of them. Then we introduce an ε-conjugate dual problem for the vector set-valued optimization problem. Finally, we establish a weak duality theorem and a strong duality theorem for the relationship between the primal and the dual problems.

In this paper, a full Nesterov–Todd-step infeasible interior-point algorithm is presented for semidefinite optimization (SDO) problems. In contrast of some classical interior-point algorithms for SDO problems, this algorithm does not need to perform computationally expensive calculations for centering steps which are needed for classical interior-point methods. The convergence analysis of the algorithm is shown and it is also proved that the complexity bound of the algorithm coincides with the currently best iteration bound obtained by infeasible interior-point algorithms for this class of optimization problems.

Optimization problems are used to model many real-life problems. Therefore, solving these problems is one of the most important goals of algorithm design. A general optimization problem can be defined by specifying a set of constraints that defines a subset in some underlying space (like the Euclidean space) called the feasible subset and an objective function that we are trying to maximize or minimize, as the case may be, over the feasible set. The difficulty of solving such problems typically depends on how ‘complex’ the feasible set and the objective function are. For example, a very important class of optimization problems is linear programming. Here the feasible subset is specified by a set of linear inequalities (in the Euclidean space); the objective function is also linear. A more general class of optimization problems is convex programming, where the feasible set is a convex subset of a Euclidean space and the objective function is also convex. Convex programs (and hence, linear programs) have a nice property that any local optimum is also a global optimum for the objective function. There are a variety of techniques for solving such problems – all of them try to approach a local optimum (which we know would be a global optimum as well). These notions are discussed in greater detail in a later section in this chapter. The more general problem, the so-called non-convex programs, where the objective function and the feasible subset could be arbitrary can be very challenging to solve. In particular, discrete optimization problems, where the feasible subset could be a (large) discrete subset of points falls under this category.

In connection with the remarkable and rapid development of linear and nonlinear vector optimization, an increasing number of papers were devoted to the construction and investigation of dual problems. Several concepts of duality were treated, especially Lagrange duality and Fenchel duality (cf. Elster /1/, Elster/Iwanow /3/, Jahn /8/). Apart of linear problems we can find extensive investigations of convex problems and more and more of nonconvex problems. In the last case some difficulties can arise concerning p.e. separation theorems which are sufficiently general. In the present paper a direct duality theorem is derived for rather general (nonconvex) vector optimization problems. For the proof a separation theorem is needed where the separating functional is nonlinear (more precisely: convex). The results are formulated in finite dimensional spaces, but an extension to infinite dimensions is possible.

Decomposition of large matrix inequalities for matrices with chordal sparsity graph has been recently used by Kojima et al. (Math Program 129(1):33–68, 2011) to reduce problem size of large scale semidefinite optimization (SDO) problems and thus increase efficiency of standard SDO software. A by-product of such a decomposition is the introduction of new dense small-size matrix variables. We will show that for arrow type matrices satisfying suitable assumptions, the additional matrix variables have rank one and can thus be replaced by vector variables of the same dimensions. This leads to significant improvement in efficiency of standard SDO software. We will apply this idea to the problem of topology optimization formulated as a large scale linear semidefinite optimization problem. Numerical examples will demonstrate tremendous speed-up in the solution of the decomposed problems, as compared to the original large scale problem. In our numerical example the decomposed problems exhibit linear growth in complexity, compared to the more than cubic growth in the original problem formulation. We will also give a connection of our approach to the standard theory of domain decomposition and show that the additional vector variables are outcomes of the corresponding discrete Steklov–Poincaré operators.

In the last three decades, the theory of variational analysis provides very effective and powerful techniques for studying a wide class of problems arising in nonlinear equations, optimization problems, economics equilibrium, game theory, complementarity problems, and fixed point problems, as well as other branches of mathematics and engineering sciences. So, the thorough study of both theory and methods about variational inequalities will help us to find new techniques for solving the practical problem. Vector optimization problems have received much attention by many authors due to their extensive applications in many fields such as biology, economics, optimal control, and differential inclusions. Because of the importance and active impact of the variational inequality and the vector optimization problem in the nonlinear analysis and optimization, this special issue, focusing on most recent contributions, includes works on variational inequalities, equilibrium problems and nonexpansive mappings, vector optimization problems and generalized convex functions, robust optimization problems, and optimization problems with applications, which are based on a strict international peer review procedure and our original proposal. A brief review of the papers is given under the following four topics.

In this paper, we consider adjustable robust versions of convex optimization problems with uncertain constraints and objectives and show that under fairly general assumptions, a static robust solution provides a good approximation for these adjustable robust problems. An adjustable robust optimization problem is usually intractable since it requires to compute a solution for all possible realizations of uncertain parameters, while an optimal static solution can be computed efficiently in most cases if the corresponding deterministic problem is tractable. The performance of the optimal static robust solution is related to a fundamental geometric property, namely, the symmetry of the uncertainty set. Our work allows for the constraint and objective function coefficients to be uncertain and for the constraints and objective functions to be convex, thereby providing significant extensions of the results in Bertsimas and Goyal (Math Oper Res 35:284–305, 2010) and Bertsimas et al. (Math Oper Res 36: 24–54, 2011b) where only linear objective and linear constraints were considered. The models in this paper encompass a wide variety of problems in revenue management, resource allocation under uncertainty, scheduling problems with uncertain processing times, semidefinite optimization among many others. To the best of our knowledge, these are the first approximation bounds for adjustable robust convex optimization problems in such generality.

Nonlinear optimization algorithms are rarely discussed from a complexity point of view. Even the concept of solving nonlinear problems on digital computers is not well defined. The focus here is on a complexity approach for designing and analyzing algorithms for nonlinear optimization problems providing optimal solutions with prespecified accuracy in the solution space. We delineate the complexity status of convex problems over network constraints, dual of flow constraints, dual of multi-commodity, constraints defined by a submodular rank function (a generalized allocation problem), tree networks, diagonal dominant matrices, and nonlinear Knapsack problem's constraint. All these problems, except for the latter in integers, have polynomial time algorithms which may be viewed within a unifying framework of a proximity-scaling technique or a threshold technique. The complexity of many of these algorithms is furthermore best possible in that it matches lower bounds on the complexity of the respective problems. In general nonseparable optimization problems are shown to be considerably more difficult than separable problems. We compare the complexity of continuous versus discrete nonlinear problems and list some major open problems in the area of nonlinear optimization.

Notes on Computational Complexity of GE Inequalities

Summary For highly nonlinear problems, the objective function f(x) may have multiple local optima and it is desired to locate all of them. Analytical or adjoint-based derivatives may not be available for most real optimization problems, especially, when responses of a system are predicted by numerical simulations. The distributed-Gauss-Newton (DGN) optimization method performs quite efficiently and robustly for history-matching problems with multiple best matches. However, this method is not applicable for generic optimization problems, e.g., life-cycle production optimization or well location optimization. In this paper, we generalized the distribution techniques of the DGN optimization method and developed a new distributed quasi-Newton (DQN) optimization method that is applicable to generic optimization problems. It can handle generalized objective functions F(x,y(x))=f(x) with both explicit variables x and implicit variables, i.e., simulated responses, y(x). The partial derivatives of F(x,y) with respect to both x and y can be computed analytically, whereas the partial derivatives of y(x) with respect to x (sensitivity matrix) is estimated by applying the same efficient information sharing mechanism implemented in the DGN optimization method. An ensemble of quasi-Newton optimization tasks is distributed among multiple high-performance-computing (HPC) cluster nodes. The simulation results generated from one optimization task are shared with others by updating a common set of training data points, which records simulated responses of all simulation jobs. The sensitivity matrix at the current best solution of each optimization task is approximated by either the linear-interpolation (LI) method or the support-vector-regression (SVR) method, using some or all training data points. The gradient of the objective function is then analytically computed using its partial derivatives with respect to x and y and the estimated sensitivities of y with respect to x. The Hessian is updated using the quasi-Newton formulation. A new search point for each distributed optimization task is generated by solving a quasi-Newton trust-region subproblem for the next iteration. The proposed DQN method is first validated on a synthetic history matching problem and its performance is found to be comparable with the DGN optimizer. Then, the DQN method is tested on different optimization problems. For all test problems, the DQN method can find multiple optima of the objective function with reasonably small numbers of iterations (30 to 50). Compared to sequential model-based derivative-free optimization methods, the DQN method can reduce the computational cost, in terms of the number of simulations required for convergence, by a factor of 3 to 10.

Abstract. In this paper, we consider and study a new class of general-ized vector quasi-variational type inequalities and obtain some existencetheorems for both under compact and noncompact assumptions in topo-logical vector spaces without using monotonicity. For the noncompactcase, we use the concept of escaping sequences. 1. IntroductionThe vector variational inequality problem was initiated by Giannessi [11] in nite dimensional Euclidean spaces with applications. Later on many authors[6, 7, 12, 15, 24] generalized vector variational inequalities in abstract spacesin several ways. The vector variational-like inequality is one of the generalizedforms of vector variational inequalities [11]. In 1989, Parida et al [19] stud-ied the existence of solutions for variational-like inequalities in R n space andhave shown a relationship between variational like inequalities problems andconvex programming as well as with complementarity problems. The vectorvariational like inequality and generalized vector variational like inequality arepowerful tools to study non-convex vector optimization problems and convexand nondi erentiable vector optimization problems respectively, see [10, 18, 21].In 1973, Bensoussan and Lions [3] introduced the quasi-variational inequalityproblem (QVIP). Since then, many generalizations of the QVIP have beenappeared in the literature (see, for example, [1, 2]). In the last decade, becauseof applications in the optimization problems, mathematical programming andequilibrium problems, the QVIP has been intensively studied by many authors[5, 8, 16, 20, 22, 23, 25].In this paper, we consider the generalized vector quasivariational type in-equality problem (GVQVTIP) and obtain some existence theorems for solutions

Robust counterparts and robust efficient solutions in vector optimization under uncertainty

This chapter presents a newly defined novel combinatorial optimisation problem, namely, the General Combinatorial Optimisation Problem (GCOP), whose decision variables are a set of elementary algorithm components. The combinations of these algorithm components, i.e. solutions of GCOP, thus represent different search algorithms. The objective of GCOP is to find the optimal combinations of algorithm components for solving optimisation problems. Solving the GCOP is thus equivalent to automatically designing the best search algorithms for optimisation problems. The definition of the GCOP is presented with a new taxonomy which categorises relevant literature on automated algorithm design into three lines of research, namely, automated algorithm configuration, selection and composition. Based on the decision space under consideration, the algorithm design itself is defined as an optimisation problem. Relevant literature is briefly reviewed, motivating a new line of exciting and challenging directions on the emerging research of automated algorithm design.

It is possible to solve unbounded convex vector optimization problems (CVOPs) in two phases: (1) computing or approximating the recession cone of the upper image and (2) solving the equivalent bounded CVOP where the ordering cone is extended based on the first phase. In this paper, we consider unbounded CVOPs and propose an alternative solution methodology to compute or approximate the recession cone of the upper image. In particular, we relate the dual of the recession cone with the Lagrange dual of weighted sum scalarization problems whenever the dual problem can be written explicitly. Computing this set requires solving a convex (or polyhedral) projection problem. We show that this methodology can be applied to semidefinite, quadratic, and linear vector optimization problems and provide some numerical examples.