Nonconvex Constrained Optimization Problems Research Articles

Iterative Algorithm for Triple-Hierarchical Constrained Nonconvex Optimization Problem and Its Application to Network Bandwidth Allocation

In this paper, we discuss a variational inequality problem with the gradient of a nonconvex objective function in which the constraint set composed of the absolute set and the subsidiary sets is not feasible. We formulate a compromise solution of the problem by using a solution to a variational inequality for the gradient over a subset of the absolute set with the elements closest to the subsidiary sets in terms of the norm. The objective of this paper is to enable us to discuss a network bandwidth allocation problem with a nonconcave utility function in which the compoundable constraints about the preferable transmission rate fall in the infeasible region. For such a bandwidth allocation, the ideal bandwidth must be quickly shared among traffic sources so as to satisfy the capacity constraints (absolute constraints) and all sources' demands regarding the transmission rate (subsidiary constraints) as much as possible. Iterative algorithms for the bandwidth allocation should thus not involve auxiliary optimization problems or complicated computations. Here, we devise an algorithm based on an iterative technique for triple-hierarchical constrained nonconvex optimization to achieve the optimal bandwidth allocation. Numerical examples for the bandwidth allocation demonstrate the effectiveness and convergence of the proposed algorithm.

Optimal Joint Water-Filling for Coordinated Transmission over Frequency-Selective Fading Channels

we consider a novel power allocation scheme for a frequency-selective fading channel with two coordinated transmission points (CTP) transmission. In order to maximize the channel's throughput, closed form solutions are obtained by solving a non-convex constrained optimization problem. The solution which we name it joint water-filling (Jo-WF) turns out to take the form of traditional WF and also combined with regular cooperative feature. Numerical results are presented to verify the optimality of the derived scheme and to show throughput gains over traditional non-coordinated WF and equal power allocation (EPA).

Pricing surplus server capacity for mean waiting time sensitive customers

We consider a queueing model wherein the resource is shared by two different classes of customers, primary (existing) and secondary (new), under a service level based pricing contract. This contract between secondary class customers and resource manager specifies unit admission price and quality of service (QoS) offered. We assume that the secondary customers’ Poisson arrival rate depends linearly on unit price and service level offered while the server uses a delay dependent priority queue management scheme. We analyze the joint problem of optimal pricing and operation of the resource with the inclusion of secondary class customers, while continuing to offer a pre-specified QoS to primary class customers. Our analysis leads to an algorithm that finds, in closed form expressions, the optimal points of the resulting non-convex constrained optimization problem. We also study in detail the structure and the non-linear nature of these optimal pricing and operating decisions.

An Improved Particle Swarm Optimization for Nonconvex Economic Dispatch Problems

This paper presents an efficient approach for solving economic dispatch (ED) problems with nonconvex cost functions using an improved particle swarm optimization (IPSO). Although the particle swarm optimization (PSO) approaches have several advantages suitable to heavily constrained nonconvex optimization problems, they still can have the drawbacks such as local optimal trapping due to premature convergence (i.e., exploration problem), insufficient capability to find nearby extreme points (i.e., exploitation problem), and lack of efficient mechanism to treat the constraints (i.e., constraint handling problem). This paper proposes an improved PSO framework employing chaotic sequences combined with the conventional linearly decreasing inertia weights and adopting a crossover operation scheme to increase both exploration and exploitation capability of the PSO. In addition, an effective constraint handling framework is employed for considering equality and inequality constraints. The proposed IPSO is applied to three different nonconvex ED problems with valve-point effects, prohibited operating zones with ramp rate limits as well as transmission network losses, and multi-fuels with valve-point effects. Additionally, it is applied to the large-scale power system of Korea. Also, the results are compared with those of the state-of-the-art methods.

Robust PID controller tuning based on the heuristic Kalman algorithm

This paper presents a simple but effective tuning strategy for robust PID controllers satisfying multiple H ∞ performance criteria. Finding such a controller is known to be computationally intractable via the conventional techniques. This is mainly due to the non-convexity of the resulting control problem which is of the fixed order/structure type. To solve this kind of control problem easily and directly, without using any complicated mathematical manipulations and without using too many “user defined” parameters, we utilize the heuristic Kalman algorithm (HKA) for the resolution of the underlying constrained non-convex optimization problem. The resulting tuning strategy is applicable both to stable and unstable systems, without any limitation concerning the order of the process to be controlled. Various numerical studies are conducted to demonstrate the validity of the proposed tuning procedure. Comparisons with previously published works are also given.

Solutions and optimality criteria for nonconvex constrained global optimization problems with connections between canonical and Lagrangian duality

This paper presents a canonical duality theory for solving a general nonconvex quadratic minimization problem with nonconvex constraints. By using the canonical dual transformation developed by the first author, the nonconvex primal problem can be converted into a canonical dual problem with zero duality gap. A general analytical solution form is obtained. Both global and local extrema of the nonconvex problem can be identified by the triality theory associated with the canonical duality theory. Illustrative applications to quadratic minimization with multiple quadratic constraints, box/integer constraints, and general nonconvex polynomial constraints are discussed, along with insightful connections to classical Lagrangian duality. Criteria for the existence and uniqueness of optimal solutions are presented. Several numerical examples are provided.

Mixed H2/H∞ residual generator design via Heuristic Kalman Algorithm

This paper presents a simple but effective synthesis strategy for observers based faults detection in linear time-invariant (LTI) systems which are simultaneously affected by two classes of unknown inputs: Noises having fixed spectral densities and unknown finite energy disturbances. The problem of designing such an observer, also called a residual generator, is formulated as a mixed H2/H∞ optimization problem. This is done to obtain an optimal residual generator, i.e. with minimal sensitivity to unknown inputs. Unfortunately, there is no known solution to this difficult optimization problem. Finding such a residual generator is known to be computationally intractable via the conventional techniques. This is mainly due to the non-convexity of the resulting optimization problem. To solve this kind of problem easily and directly, without using any complicated mathematical manipulations, we utilize the Heuristic Kalman Algorithm (HKA) for the resolution of the underlying constrained non-convex optimization problem. A numerical example is given to illustrate the advantage of the mixed H2/H∞ optimization approach against techniques based on optimization of H2 or H∞ criteria.

MV-PURE Estimator: Minimum-Variance Pseudo-Unbiased Reduced-Rank Estimator for Linearly Constrained Ill-Conditioned Inverse Problems

This paper proposes a novel estimator named minimum-variance pseudo-unbiased reduced-rank estimator (MV- PURE) for the linear regression model, designed specially for the case where the model matrix is ill-conditioned and the unknown deterministic parameter vector to be estimated is subjected to known linear constraints. As a natural generalization of the Gauss-Markov (BLUE) estimator, the MV-PURE estimator is a solution of the following hierarchical nonconvex constrained optimization problem directly related to the mean square error expression. In the first-stage optimization, under a rank constraint, we minimize simultaneously all unitarily invariant norms of an operator applied to the unknown parameter vector in view of suppressing bias of the proposed estimator. Then, in the second-stage optimization, among all pseudo-unbiased reduced-rank estimators defined as the solutions of the first-stage optimization, we find the one achieving minimum variance. We derive a closed algebraic form of the MV-PURE estimator and show that well-known estimators-the Gauss-Markov (BLUE) estimator, the generalized Marquardt's reduced-rank estimator, and the minimum-variance conditionally unbiased affine estimator subject to linear restrictions-are all special cases of the MV-PURE estimator. We demonstrate the effectiveness of the proposed estimator in a numerical example, where we employ the MV-PURE estimator to the ill-conditioned problem of reconstructing a 2-D image subjected to linear constraints from blurred, noisy observation. This example demonstrates that the MV-PURE estimator outperforms all aforementioned estimators, as it achieves smaller mean square error for all values of signal-to-noise ratio.

Investigating a hybrid simulated annealing and local search algorithm for constrained optimization

Constrained Optimization Problems (COP) often take place in many practical applications such as kinematics, chemical process optimization, power systems and so on. These problems are challenging in terms of identifying feasible solutions when constraints are non-linear and non-convex. Therefore, finding the location of the global optimum in the non-convex COP is more difficult as compared to non-convex bound-constrained global optimization problems. This paper proposes a Hybrid Simulated Annealing method ( HSA), for solving the general COP. HSA has features that address both feasibility and optimality issues and here, it is supported by a local search procedure, Feasible Sequential Quadratic Programming ( FSQP). We develop two versions of HSA. The first version ( HSAP) incorporates penalty methods for constraint handling and the second one ( HSAD) eliminates the need for imposing penalties in the objective function by tracing feasible and infeasible solution sequences independently. Numerical experiments show that the second version is more reliable in the worst case performance.

Existence of Exact Penalty and Its Stability for Nonconvex Constrained Optimization Problems in Banach Spaces

In this paper we use the penalty approach in order to study two constrained minimization problems. A penalty function is said to have the generalized exact penalty property if there is a penalty coefficient for which approximate solutions of the unconstrained penalized problem are close enough to approximate solutions of the corresponding constrained problem. In this paper we show that the generalized exact penalty property is stable under perturbations of cost functions, constraint functions and the right-hand side of constraints.

Intensity modulated radiotherapy treatment planning by use of a barrier-penalty multiplier method

The use of nonlinear functions describing biological effects has recently become a major goal in connection with intensity modulated radiotherapy planning models for cancer treatment. In this article, we present a new biological model for this purpose and discuss the solution of the related large-scale nonlinear optimization problems. The model includes equivalent uniform dose and partial volume constraints and employs tumor control probability as the objective. The resulting optimization problems are convex; there are nonconvex constrained optimization problems with several thousands of variables for which gradients of the involved functions are available, but the computation of Hessians is too costly. It is suggested to solve these problems using the barrier-penalty multiplier method by Polyak ([Polyak, R., 1992, Modified barrier functions (theory and methods). Mathematical Programming, 54, 177–222.], [Polyak, R., 2002, Nonlinear rescaling vs. smoothing technique in convex optimization. Mathematical Programming, 92, 197–235.]) and Ben-Tal et al. and Ben-Tal and Zibulevsky ([Ben-Tal, A., Yuzefokich, I. and Zibulevsky, M., 1992, Penalty/barrier multiplier methods for minimax and constrained smooth convex problems. Technical Report 9/92, Optimization Laboratory, Faculty of Industrial Engineering and Management, Technion, Haifa, Israel.], [Ben-Tal, A. and Zibulevsky, M., 1997, Penalty/barrier multiplier methods for convex programming problems. SIAM Journal of Optimization, 7, 347–366.]), where this algorithm is modified according to ideas which are motivated by the related Lagrangian barrier algorithm of Conn et al. ([Conn, A.R., Gould, N.I.M. and Taint, P., 1992, A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds. Technical Report 92/07, Department of Maths, FUNDP, Namur, Belgium.], [Conn, A.R., Gould, N.I.M. and Taint, P.L., 1992, A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds. Mathematics of Computation, 66, 261–288.]). In particular, the subproblems in the algorithm are solved by a conjugate gradient method, as the spectrum of the Hessian of the Lagrangian at a solution of such problem indicates fast (local) convergence of the objective function values to a good approximate (locally) optimal value. Some characteristic numbers showing the average numerical performance of the algorithm are tabulated for various types of tumors and for a set of 127 clinical cases in total. Its capabilities and typical behavior also are illustrated explicitly by a computed therapy plan for a difficult clinical case of a larynx tumor.

A geometric framework for nonconvex optimization duality using augmented lagrangian functions

We provide a unifying geometric framework for the analysis of general classes of duality schemes and penalty methods for nonconvex constrained optimization problems. We present a separation result for nonconvex sets via general concave surfaces. We use this separation result to provide necessary and sufficient conditions for establishing strong duality between geometric primal and dual problems. Using the primal function of a constrained optimization problem, we apply our results both in the analysis of duality schemes constructed using augmented Lagrangian functions, and in establishing necessary and sufficient conditions for the convergence of penalty methods.

Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions

In this paper, we first examine how global optimality of non-convex constrained optimization problems is related to Lagrange multiplier conditions. We then establish Lagrange multiplier conditions for global optimality of general quadratic minimization problems with quadratic constraints. We also obtain necessary global optimality conditions, which are different from the Lagrange multiplier conditions for special classes of quadratic optimization problems. These classes include weighted least squares with ellipsoidal constraints, and quadratic minimization with binary constraints. We discuss examples which demonstrate that our optimality conditions can effectively be used for identifying global minimizers of certain multi-extremal non-convex quadratic optimization problems.

Structural optimization of forest machines with hybridized nonsmooth and global methods

We consider an optimal shape design problem of grapple loaders. Our aim is to minimize the weight of the machine. The durability is taken into account via stress and buckling constraints. Thus we have a nonlinear and nonconvex constrained optimization problem, which, due to the buckling constraints, is also nondifferentiable. The optimization is realized by using the hybridization of a genetic algorithm and a nonsmooth proximal bundle method in order to maintain the advantages of both the methods: reliability and computational efficiency, respectively.

Existence of a Saddle Point in Nonconvex Constrained Optimization

The existence of a saddle point in nonconvex constrained optimization problems is considered in this paper. We show that, under some mild conditions, the existence of a saddle point can be ensured in an equivalent p-th power formulation for a general class of nonconvex constrained optimization problems. This result expands considerably the class of optimization problems where a saddle point exists and thus enlarges the family of nonconvex problems that can be solved by dual-search methods.

Necessary Optimality Conditions for Nonsmooth Multicriterial Optimization Problems

Necessary optimality conditions of the first order are derived for nonsmooth non-convex constrained optimization problems where the cost mapping is vector-valued and all occurring spaces are infinite-dimensional. These necessary conditions are given in the Karush–Kuhn–Tucker formulation and hold for various optimality concepts as proper and weak efficiency. An investigation and comparison of different constraint qualifications is also included.

Solutions for a class of invex programming problems

A nonconvex constrained optimization problem is considered in which the constraints are of the form of generalized polynomials. An invexity kernel is established for this class of problem, and a consequent theorem gives sufficient conditions for the solutions of such problems.

Semismooth and Semiconvex Functions in Constrained Optimization

We introduce semismooth and semiconvex functions and discuss their properties with respect to nonsmooth nonconvex constrained optimization problems. These functions are locally Lipschitz, and hence have generalized gradients. The author has given an optimization algorithm that uses generalized gradients of the problem functions and converges to stationary points if the functions are semismooth. If the functions are semiconvex and a constraint qualification is satisfied, then we show that a stationary point is an optimal point. We show that the pointwise maximum or minimum over a compact family of continuously differentiable functions is a semismooth function and that the pointwise maximum over a compact family of semiconvex functions is a semiconvex function. Furthermore, we show that a semismooth composition of semismooth functions is semismooth and give a type of chain rule for generalized gradients.