Nonsmooth Programming Research Articles

Necessary optimality conditions for strictly robust bilevel optimization problems

The study of robust bilevel programming problems is a relatively new area of optimization theory. In this work, we investigate a bilevel optimization problem where the upper-level and the lower-level constraints incorporate uncertainty. Reducing the problem into a single-level nonlinear and nonsmooth program, necessary optimality conditions are then developed in terms of Clarke subdifferentials. Our approach consists of using the optimal value reformulation together with a partial calmness condition for the robust counterpart of the initial problem. To aid in the detection of Karush-Kuhn-Tucker (KKT) multipliers, an appropriate nonsmooth Mangasarian-Fromovitz constraint qualification is introduced. There are examples highlighting both our results and the limits of certain past studies.

On Frèchet normal cone for nonsmooth mathematical programming problems with switching constraints

This paper is devoted to the study of a class of nonsmooth programming problems with switching constraints (abbreviated as, (NMPSC)), where all the involved functions in the switching constraints are assumed to be locally Lipschitz. We investigate the properties of Frèchet normal cone of (NMPSC). In particular, we introduce two Guignard type constraint qualifications for (NMPSC) in terms of Michel-Penot subdifferential. Moreover, we derive two estimates for the Frèchet normal cone of (NMPSC) and further establish stationarity conditions at an optimal solution for (NMPSC). To the best of our knowledge, this is for the first time Frèchet normal cone for (NMPSC) have been studied in the setting of Euclidean spaces.

Chaotic heterogeneous comprehensive learning PSO method for size and shape optimization of structures

This paper proposes a novel chaotic heterogeneous comprehensive learning particle swarm optimization (CLPSO) method for the simultaneous size and shape design of structures. The heterogeneous CLPSO divides the particles into two explorative and exploitative subpopulations. The exploration performs the global searches for the set of best particles experienced solely within its own subpopulation, whilst the exploitation refines the deep searches learnt from the global best particle over an entire population. In essence, the proposed method maintains a good balance between the global explorative and local exploitative optimization schemes. The global searches within the explorative subpopulation are independent to the exploitative simulations even if the latter scheme prematurely converges to the local swarm position. For both subpopulations, the comprehensive learning approach constructs the particles through a cross-positioning process on the individual variable space and avoids the local optimal pitfall. Moreover, the chaotic logistic map within the exploitative optimization tests the global best particle through the set of diversely generated samples and hence enhances the local search ability. Various enriching techniques, including automatic adaptive (inertial weight and acceleration) parameters with dynamic space reduction, are incorporated to improve the likelihood of finding the optimal solution of practical-scale problems at modest computing efforts. The accuracy and robustness of the proposed method are illustrated through a number of planar and spatial truss design benchmarks subjected to the challenging nonconvex and/or nonsmooth programs.

On interval-valued bilevel optimization problems using upper convexificators

In this paper, we investigate a bilevel interval valued optimization problem. Reducing the problem into a one-level nonlinear and nonsmooth program, necessary optimality conditions are developed in terms of upper convexificators. Our approach consists of using an Abadie’s constraint qualification together with an appropriate optimal value reformulation. Later on, using an upper estimate for upper convexificators of the optimal value function, we give a more detailed result in terms of the initial data. The appearing functions are not necessarily Lipschitz continuous, and neither the objective function nor the constraint functions of the lower-level optimization problem are assumed to be convex. There are additional examples highlighting both our results and the limitations of certain past studies.

Stochastic Difference-of-Convex-Functions Algorithms for Nonconvex Programming

The paper deals with stochastic difference-of-convex-functions (DC) programs, that is, optimization problems whose cost function is a sum of a lower semicontinuous DC function and the expectation of a stochastic DC function with respect to a probability distribution. This class of nonsmooth and nonconvex stochastic optimization problems plays a central role in many practical applications. Although there are many contributions in the context of convex and/or smooth stochastic optimization, algorithms dealing with nonconvex and nonsmooth programs remain rare. In deterministic optimization literature, the DC algorithm (DCA) is recognized to be one of the few algorithms able to effectively solve nonconvex and nonsmooth optimization problems. The main purpose of this paper is to present some new stochastic DCAs for solving stochastic DC programs. The convergence analysis of the proposed algorithms is carefully studied, and numerical experiments are conducted to justify the algorithms' behaviors.

On the 𝑞-analogue of the Pair Correlation Conjecture via Fourier optimization

We study the q q -analogue of the average of Montgomery’s function F ( α , T ) F(\alpha ,\, T) over bounded intervals. Assuming the Generalized Riemann Hypothesis for Dirichlet L L -functions, we obtain upper and lower bounds for this average over an interval that are quite close to the pointwise conjectured value of 1 1 . To compute our bounds, we extend a Fourier analysis approach by Carneiro, Chandee, Chirre, and Milinovich, and apply computational methods of non-smooth programming.

Constraint qualifications in terms of convexificators for nonsmooth programming problems with mixed constraints

The main aim of the paper is to introduce certain constraint qualifications for a nonsmooth programming problem in terms of semi-regular convexificators and investigate their relations with other existing notions of constraint qualifications. The programming problem under consideration has mixed constraints, that is, it involves both inequality and equality constraints. All these notions are in terms of upper semi-regular convexificators of inequality constraints and pseudo-differentials of equality constraints. Based on a sufficient condition for error bound property, the implication relation between quasinormality and error bound property in terms of convexificators is investigated in this paper. Three conditions are introduced, namely constant positive linear dependence condition (CPLD), constant rank constraint qualification (CRCQ) and Mangasarian–Fromovitz constraint qualification (MFCQ) in terms of convexificators. These conditions are in fact shown to be constraint qualifications as Karush–Kuhn–Tucker optimality conditions hold when CPLD holds and both MFCQ and CRCQ imply CPLD. Further, it is observed that CPLD and quasinormality conditions are independent for nonsmooth problems in terms of convexificators.

Coded Caching With Device Computing in Mobile Edge Computing Systems

Edge caching and computing have been regarded as an efficient approach to tackle the wireless spectrum crunch problem. In this paper, we design a general coded caching with device computing strategy for computational tasks, e.g., virtual reality (VR) rendering, to minimize the average transmission bandwidth under the quality of service guarantee. Because both coded data and stored data can be the data before or after computing, the proposed scheme has numerous edge computing and caching paths corresponding to different bandwidth requirement. We thus formulate a joint coded caching and computing optimization problem to decide whether a mobile device stores the data before computing or the data after computing, which tasks to be coded cached and which tasks to be computed locally. The optimization problem is shown to be 0–1 non-convex non-smooth programming, and can be decomposed into a computation offloading programming and a coded caching programming. For a computation offloading subproblem, we proposed an algorithm which applies the convergence of the alternating direction method of multipliers (ADMM) under a non-convex programming and the wide application of the concave-convex procedure (CCCP) for difference of convex (DC) programming to obtain a stationary point, and numerical results verify the convergence of the proposed algorithm and its suboptimality. For the coded cache programming, we design a low complexity algorithm to obtain an acceptable solution. Numerical results demonstrate that the proposed scheme provides a significant bandwidth saving by taking full advantage of the caching and computing capability of mobile devices.

Formula omitted]-[formula omitted]-invexity in [formula omitted]-differentiable mathematical programming

In this paper, a new concept of generalized convexity is introduced for (not necessarily) differentiable optimization problem with E-differentiable functions. Namely, for an E-differentiable function, the concept of E-B-invexity is defined. The E-differentiable E-B-invexity notion unify the concepts of convexity, invexity, E-convexity, E-invexity and B-invexity. Further, the sufficiency of the so-called E-Karush-Kuhn–Tucker optimality conditions are established for the considered E-differentiable optimization problem with both inequality and equality constraints under E-B-invexity hypotheses. Moreover, the example of a nonsmooth programming problem with E-differentiable functions is constructed to illustrate the aforesaid results.

A Redistributed Bundle Algorithm Based on Local Convexification Models for Nonlinear Nonsmooth DC Programming

Abstract For nonlinear nonsmooth DC programming (difference of convex functions), we introduce a new redistributed proximal bundle method. The subgradient information of both the DC components is gathered from some neighbourhood of the current stability center and it is used to build separately an approximation for each component in the DC representation. Especially we employ the nonlinear redistributed technique to model the second component of DC function by constructing a local convexification cutting plane. The corresponding convexification parameter is adjusted dynamically and is taken sufficiently large to make the ”augmented” linearization errors nonnegative. Based on above techniques we obtain a new convex cutting plane model of the original objective function. Based on this new approximation the redistributed proximal bundle method is designed and the convergence of the proposed algorithm to a Clarke stationary point is proved. A simple numerical experiment is given to show the validity of the presented algorithm.

New optimality conditions for bilevel programs by using an exact separation principle

In this paper, we are concerned with a bilevel optimization problem $$\left( P\right) $$ . Reducing it into a one-level nonlinear and nonsmooth program using an exact separation theorem we give necessary optimality conditions in terms of Frechet subdifferentials and Frechet normal cones.

Constraint qualifications for nonsmooth programming

ABSTRACTBy using an implicit function theorem and a result of error bound, we provide new constraint qualifications ensuring the Karush–Kuhn–Tuker necessary optimality conditions for both smooth and nonsmooth optimization problems in normed spaces or Banach spaces.

Joint multicast beamforming and user scheduling in large‐scale antenna systems

This paper studies the joint multicast beamforming and user scheduling problem, with the objective of minimizing total transmitting power across multiple channels by jointly assigning each user to appropriate channel and designing multicast beamformer for each channel. The problem of interest is formulated in two different optimization problems, a mixed binary quadratically constrained quadratic program and a highly-structured nonsmooth program. Two different algorithms, based on convex relaxation and convex restriction, respectively, are proposed to solve the problem. The performance ratio between the approximate solution provided by the convex-relaxation-based algorithm and optimal solution is proved to be upper bounded by a constant independent of problem data. The convex-restriction-based algorithm is guaranteed to converge to a critical point to the nonsmooth formulation problem. Finally, extensive simulation results verify the theoretical analysis and demonstrate the advantage of the proposed co-design scheme over conventional fixed scheduling and random scheduling in terms of power consumption.

The Optimization of the Mean-Variance Portfolio Selection with Nonsmooth Concave Transaction Costs

Transaction cost was an important factor in the financial market. According to the situation of the underdeveloped market, we proposed a mean-variance portfolio selection model with nonsmooth concave transaction costs. This is nonsmooth programming problem. The paper proposed the subsection method and pivoting algorithm to solve it, and proved these algorithms convergent. Finally, according to annual yields of six kinds of securities in eight years, we solved the mean-variance portfolio optimal strategy with nonsmooth concave transaction costs. The example was given to illustrate the efficiency of the algorithms, which offered a new way to solve the problems of nonsmooth programming.

Optimality Conditions for Bilevel Vector Optimization Problems with a Variable Ordering Structure

ABSTRACTThe article is concerned with the optimistic formulation of a multiobjective bilevel optimization problem with locally Lipschitz continuous inclusion constraints. Using a variable ordering structure defined by a Bishop–Phelps cone, we investigate necessary optimality conditions for locally weakly nondominated solutions. Reducing the problem into a one-level nonlinear and nonsmooth program, we use the extremal principle by Mordukhovich to get fuzzy optimality conditions. More explicit conditions with the initial data are obtained using both the Ekeland’s variational principle and the support function. Fortunately, the Lipschitz property of a set-valued mapping is conserved for its support function. An appropriate regularity condition is given to help us discern the Lagrange-Kuhn-Tucker multipliers.

OPTIMALITY AND PARAMETRIC DUALITY FOR NONSMOOTH MINIMAX FRACTIONAL PROGRAMMING PROBLEMS INVOLVING L-INVEX-INFINE FUNCTIONS

The Karush-Kuhn-Tucker type necessary optimality conditions are given for the nonsmooth minimax fractional programming problem with inequality and equality constraints. Subsequently, based on the idea of L-invex-infine functions defined in terms of the limiting/Mordukhovich subdifferential of locally Lipschitz functions, we obtain sufficient optimality conditions for the considered nonsmooth minimax fractional programming problem and also we provide an example to justify the existence of sufficient optimality conditions. Furthermore, we propose a parametric type dual problem and explore duality results.

A Sequential Convex Program Approach to an Inverse Linear Semidefinite Programming Problem

This paper is devoted to the study of solving method for a type of inverse linear semidefinite programming problem in which both the objective parameter and the right-hand side parameter of the linear semidefinite programs are required to adjust. Since such kind of inverse problem is equivalent to a mathematical program with semidefinite cone complementarity constraints which is a rather difficult problem, we reformulate it as a nonconvex semi-definte programming problem by introducing a nonsmooth partial penalty function to penalize the complementarity constraint. The penalized problem is actually a nonsmooth DC programming problem which can be solved by a sequential convex program approach. Convergence analysis of the penalty models and the sequential convex program approach are shown. Numerical results are reported to demonstrate the efficiency of our approach.

Interior proximal bundle algorithm with variable metric for nonsmooth convex symmetric cone programming

This paper is devoted to the study of a bundle proximal-type algorithm for solving the problem of minimizing a nonsmooth closed proper convex function subject to symmetric cone constraints, which include the positive orthant in , the second-order cone, and the cone of positive semidefinite symmetric matrices. On the one hand, the algorithm extends the proximal algorithm with variable metric described by Alvarez et al. to our setting. We show that the sequence generated by the proposed algorithm belongs to the interior of the feasible set by an appropriate choice of a regularization parameter. Also, it is proven that each limit point of the sequence generated by the algorithm solves the problem. On the other hand, we provide a natural extension of bundle methods for nonsmooth symmetric cone programs. We implement and test numerically our bundle algorithm with some instances of Euclidean Jordan algebras.

Smoothing and SAA method for stochastic programming problems with non-smooth objective and constraints

We consider a stochastic non-smooth programming problem with equality, inequality and abstract constraints, which is a generalization of the problem studied by Xu and Zhang (Math Program 119:371---401, 2009) where only an abstract constraint is considered. We employ a smoothing technique to deal with the non-smoothness and use the sample average approximation techniques to cope with the mathematical expectations. Then, we investigate the convergence properties of the approximation problems. We further apply the approach to solve the stochastic mathematical programs with equilibrium constraints. In addition, we give an illustrative example in economics to show the applicability of proposed approach.

Generalized Convexity and Characterization of (Weak) Pareto-Optimality in Nonsmooth Multiobjective Optimization Problems

Efforts to characterize optimality in nonsmooth and/or nonconvex optimization problems have made rapid progress in the past four decades. Nonsmooth analysis, which refers to differential analysis in the absence of differentiability, has grown rapidly in recent years, and plays a vital role in functional analysis, information technology, optimization, mechanics, differential equations, decision making, etc. Furthermore, convexity has been increasingly important nowadays in the study of many pure and applied mathematical problems. In this paper, some new connections between three major fields, nonsmooth analysis, convex analysis, and optimization, are provided that will help to make these fields accessible to a wider audience. In this paper, at first, we address some newly reported and interesting applications of multiobjective optimization in Management Science and Biology. Afterwards, some sufficient conditions for characterizing the feasible and improving directions of nonsmooth multiobjective optimization problems are given, and using these results a necessary optimality condition is proved. The sufficient optimality conditions are given utilizing a generalized convexity notion. Establishing necessary and sufficient optimality conditions for nonsmooth fractional programming problems is the next aim of the paper. We follow the paper by studying (strictly) prequasiinvexity and pseudoinvexity. Finally, some connections between these notions as well as some applications of these concepts in optimization are given.