Exponential penalty function formulation for multilevel optimization using the analytical target cascading framework

In this paper, the classical exact absolute value penalty function method is used for solving a new class of nonconvex nonsmooth optimization problems. Nonconvex nondifferentiable optimization problems with both inequality and equality constraints are considered here, in which not all functions constituting them have the fundamental property of convex functions and most classes of generalized convex functions—namely, a stationary point of such a function is its global minimum. It is proved for such nonconvex optimization problems that there exists a finite threshold of penalty parameters equal to the largest absolute value of a Lagrange multiplier such that, for every penalty parameter exceeding this lower bound, there is the equivalence between an optimal solution in the original constrained minimization problem with ‐invex functions and a minimizer in its associated penalized optimization problem with the exact l1 penalty function. Further, under nondifferentiable ‐invexity assumptions, a characterization of a saddle point of the Lagrange function, defined for the considered constrained optimization problem in terms of minimizers of its associated exact penalized optimization problem with the exact l1 penalty function, is presented.

In this short paper, we study the problem of traversing a crossbar through a bent channel, which has been formulated as a nonlinear convex optimization problem. The result is a MATLAB code that we can use to compute the maximum length of the crossbar as a function of the width of the channel (its two parts) and the angle between them. In case they are perpendicular to each other, the result is expressed analytically and is closely related to the astroid curve (a hypocycloid with four cusps).

In this paper, energy efficient power control for small cells underlaying a macro cellular network is investigated. We formulate the power control problem in self-organizing small cell networks as a non-cooperative game, and propose a distributed energy efficient power control scheme, which allows the small base stations (SBSs) to take individual decisions for attaining the Nash equilibrium (NE) with minimum information exchange. Specially, in the non-cooperative power control game, a non-convex optimization problem is formulated for each SBS to maximize their energy efficiency (EE). By exploiting the properties of parameter-free fractional programming and the concept of perspective function, the non-convex optimization problem for each SBS is transformed into an equivalent constrained convex optimization problem. Then, the constrained convex optimization problem is converted into an unconstrained convex optimization problem by exploiting the mixed penalty function method. The inequality constraints are eliminated by introducing the logarithmic barrier functions and the equality constraint is eliminated by introducing the quadratic penalty function. We also theoretically show the existence and the uniqueness of the NE in the non-cooperative power control game. Simulation results show remarkable improvements in terms of EE by using the proposed scheme.

Many engineering applications face the problem of bounding the expected value of a quantity of interest (performance, risk, cost, etc.) that depends on stochastic uncertainties whose probability distribution is not known exactly. Optimal uncertainty quantification (OUQ) is a framework that aims at obtaining the best bound in these situations by explicitly incorporating available information about the distribution. Unfortunately, this often leads to non-convex optimization problems that are numerically expensive to solve. This thesis emphasizes on efficient numerical algorithms for OUQ problems. It begins by investigating several classes of OUQ problems that can be reformulated as convex optimization problems. Conditions on the objective function and information constraints under which a convex formulation exists are presented. Since the size of the optimization problem can become quite large, solutions for scaling up are also discussed. Finally, the capability of analyzing a practical system through such convex formulations is demonstrated by a numerical example of energy storage placement in power grids. When an equivalent convex formulation is unavailable, it is possible to find a convex problem that provides a meaningful bound for the original problem, also known as a convex relaxation. As an example, the thesis investigates the setting used in Hoeffding's inequality. The naive formulation requires solving a collection of non-convex polynomial optimization problems whose number grows doubly exponentially. After structures such as symmetry are exploited, it is shown that both the number and the size of the polynomial optimization problems can be reduced significantly. Each polynomial optimization problem is then bounded by its convex relaxation using sums-of-squares. These bounds are found to be tight in all the numerical examples tested in the thesis and are significantly better than Hoeffding's bounds.

A solution algorithm for non-convex mixed integer optimization problems with only few continuous variables

Adaptive modeling strategy for constrained global optimization with application to aerodynamic wing design

The optimal design problem of a conventional polyphase reluctance motor is formulated as a general Nonlinear Programming problem- The specifications like pull-out torque, temperature rise, full load power factor etc, are taken as constraints. The cost of iron and copper required for the motor is used as the objective function. The objective and constraint functions are expressed in terms of the important machine dimensions which are assumed to be continuously variable. The constrained optimization problem is convened into a sequence of unconstrained optimization problems using Zangwill's exterior penalty function formulation. For solving the unconstrained optimization problems, different methods (gradient and nongradient) are employed and their performance compared in case of a 10 HP, 2-pole, 400-volt, 50 Hz three-phase reluctance motor.

In this paper, we present a conditional gradient type (CGT) method for solving a class of composite optimization problems where the objective function consists of a (weakly) smooth term and a (strongly) convex regularization term. While including a strongly convex term in the subproblems of the classical conditional gradient method improves its rate of convergence, it does not cost per iteration as much as general proximal type algorithms. More specifically, we present a unified analysis for the CGT method in the sense that it achieves the best known rate of convergence when the weakly smooth term is nonconvex and possesses (nearly) optimal complexity if it turns out to be convex. While implementation of the CGT method requires explicitly estimating problem parameters like the level of smoothness of the first term in the objective function, we also present a few variants of this method which relax such estimation. Unlike general proximal type parameter free methods, these variants of the CGT method do not require any additional effort for computing (sub)gradients of the objective function and/or solving extra subproblems at each iteration. We then generalize these methods under stochastic setting and present a few new complexity results. To the best of our knowledge, this is the first time that such complexity results are presented for solving stochastic weakly smooth nonconvex and (strongly) convex optimization problems.

We develop new fast and efficient algorithms for designing single or multiple unimodular waveforms with good auto- and cross-correlation or weighted correlation properties, which are highly desired in radar and communication systems. The waveform design is based on the minimization of the integrated sidelobe level (ISL) and weighted ISL (WISL) of waveforms. As the corresponding problems can quickly grow to a large scale with increasing the code length and the number of waveforms, the main issue turns to be the development of fast large-scale optimization techniques. The difficulty is also that the corresponding optimization problems are nonconvex, but the required accuracy is high. Therefore, we formulate the ISL and WISL minimization problems as nonconvex quartic optimization problems in frequency domain, and then simplify them into quadratic problems via majorization-minimization technique, which is one of the basic techniques for addressing large-scale and/or nonconvex optimization problems. While designing our fast algorithms, we explore and use the inherent algebraic structures in objective functions to rewrite them into quartic forms, and in the case of WISL minimization, to derive additionally an alternative quartic form that allows us to apply the quartic-quadratic transformation. Our algorithms are applicable to large-scale unimodular waveform design problems as they are proved to have lower or comparable computational burden (analyzed theoretically) and faster convergence speed (confirmed by comprehensive simulations) than the state-of-the-art algorithms. In addition, the waveforms designed by our algorithms demonstrate better correlation properties compared to their counterparts.

A neurodynamic optimization technique based on overestimator and underestimator functions for solving a class of non-convex optimization problems

This paper gives a new subspace correction algorithm for nonlinear unconstrained convex optimization problems based on the multigrid approach proposed by S. Nash in 2000 and the subspace correction algorithm proposed by X. Tai and J. Xu in 2001. Under some reasonable assumptions, we obtain the convergence as well as a convergence rate estimate for the algorithm. Numerical results show that the algorithm is effective.

We consider minimizing a class of nonconvex composite stochastic optimization problems, and deterministic optimization problems whose objective function consists of an expectation function (or an average of finitely many smooth functions) and a weakly convex but potentially nonsmooth function. And in this paper, we focus on the theoretical properties of two types of stochastic splitting methods for solving these nonconvex optimization problems: stochastic alternating direction method of multipliers and stochastic proximal gradient descent. In particular, several inexact versions of these two types of splitting methods are studied. At each iteration, the proposed schemes inexactly solve their subproblems by using relative error criteria instead of exogenous and diminishing error rules, which allows our approaches to handle some complex regularized problems in statistics and machine learning. Under mild conditions, we obtain the convergence of the schemes and their computational complexity related to the evaluations on the component gradient of the smooth function, and find that some conclusions of their exact counterparts can be recovered.

This work aims to learn the directed acyclic graph (DAG) that captures the instantaneous dependencies underlying a multivariate time series. The observed data follow a linear structural vector autoregressive model (SVARM) with both instantaneous and time-lagged dependencies, where the instantaneous structure is modeled by a DAG to reflect potential causal relationships. While recent continuous relaxation approaches impose acyclicity through smooth constraint functions involving powers of the adjacency matrix, they lead to non-convex optimization problems that are challenging to solve. In contrast, we assume that the underlying DAG has only non-negative edge weights, and leverage this additional structure to impose acyclicity via a convex constraint. This enables us to cast the problem of non-negative DAG recovery from multivariate time-series data as a convex optimization problem in abstract form, which we solve using the method of multipliers. Crucially, the convex formulation guarantees global optimality of the solution. Finally, we assess the performance of the proposed method on synthetic time-series data, where it outperforms existing alternatives.

The various robust linear programming models investigated so far in the literature essentially appear to be based either on what is referred to as rowwise uncertainty models or on columnwise uncertainty models (these basically assume that the rows, or the columns, of the constraint matrix are subject to changes within a well-specified uncertainty set). In this chapter, we discuss a special case of columnwise uncertainty, namely the subclass of robust linear programming (LP) models with uncertainty limited to the right hand side (RHS) only (this subclass does not appear to have been significantly investigated so far). In this context we introduce the concept of a ‘two-stage robust LP model’ as opposed to the standard case (which might be referred to as a ‘single-stage robust LP model’) and we address the question of whether LP duality can be used to convert a LP problem with RHS uncertainty into a robust LP problem with uncertainty on the objective function. We show how to derive both statements of (a) the dual to the robust model and (b) the robust version of the dual. The resulting expressions of the objective function to be optimized in both cases, appear to be clearly distinct. Moreover, from a complexity point of view, one appears to be efficiently solvable (it reduces to a convex optimization problem), whereas the other, as a nonconvex optimization problem, is expected to be computationally difficult in the general case. As an application of the two-stage robust LP model introduced here, we next investigate the PERT (program evaluation and review technique) scheduling problem, considering two possible natural ways of specifying the uncertainty set for the task durations: the case where the uncertainty set is a scaled ball with respect to the L ∞ norm; the case where the uncertainty set is a scaled Hamming ball of bounded radius (which, though leading to a quite different model, bears some resemblance to the well-known Bertsimas–Sim approach to robustness). We show that in both cases the resulting robust optimization problem can be efficiently solved in polynomial time.

As for deployment of security strategy, corresponding forwarding rules for switches can be given in allusion to different traffic conditions. However, due to lack of global cognitive control for security strategy deployment in traditional Internet of Vehicles (IoV), it is quite difficult to realize global and optimized security strategy deployment scheme so as to meet security requirements in different traffic conditions. On basis of traditional IoV, cognitive engine is added in cognitive IoV (CIoV) to enhance the intelligence of traditional IoV. In allusion to CIoV, and in consideration of restrictions on transmission delay, the security strategy deployment for switches on core network is formulated in this paper, thus not only the safe transmission rules are met, but the transmission delay can also be the lowest. To be specific, the path selection of switches is modeled as 0-1 programming problem in this paper, and that optimization problem is proved to be a nonconvex optimization problem. Then we convert that problem into a convex optimization problem by log-det heuristic algorithm, thus to give path selection scheme to meet security requirements with the lowest delay on the whole. Experiment proves that cognitive engine-based security strategy deployment put forth in this paper is much better than other schemes.