A General Class of Convexification Transformation for the Noninferior Frontier of a Multiobjective Program

A general class of convexification transformations is proposed to convexify the noninferior frontier of a multiobjective program. We prove that under certain assumptions the noninferior frontier could be convexified completely or partly after transformation and then weighting method can be applied to identify the noninferior solutions. Numerical experiments are given to vindicate our results.

Data-driven Convexification for Frequency Nadir Constraint of Unit Commitment

The increasing penetration of the renewable energy sources brings new challenges to the frequency security of power systems. In order to guarantee the system frequency security, frequency constraints are incorporated into unit commitment(UC) models. Due to the non-convex form of the frequency nadir constraint which makes the frequency constrained UC(FCUC) intractable, this letter proposes a revised support vector machine(SVM) based system parameter separating plane method to convexify it. Based on this data-driven convexification method, we obtain a tractable FCUC model which is formulated as a mixed-integer quadratic programming(MIQP) problem. Case studies indicate that the proposed method can obtain less conservative solution than the existing methods with higher efficiency.

Convexification Technique for Optimal Thermal-Wind-Photovoltaic Coordination Dispatch Incorporating an Energy Storage System

This paper proposes a solution to implementing acoordinated optimal day-ahead dispatch in a hybrid thermalwind-photovoltaic power system incorporating an energy storagesystem (ESS). Our aim is to minimize total generation costand restrain the frequent change of ESS charging/dischargingstatus while meeting a series of system operating constraints,including a proposed coordinated dispatch strategy for thepurpose of reducing thermal power fluctuations. A novel twostage convexification technique (TSCT) is designed and leveragedto convert the original non-convex optimal day-ahead dispatchmodel, without taking into account the constraints of the proposed coordinated dispatch strategy into two convex quadraticprogramming problems. When introducing the constraint ofthe coordinated dispatch strategy, the corresponding inequalityconstraints are transformed into a series of linear equalityconstraints, after which the original optimal day-ahead dispatchmodel can be solved by the TSCT mentioned above. Finally,numerical simulations and comparative analysis are performedon the IEEE standard test systems to verify the validity andeffectiveness of the proposed model and method.

CONVEXIFICATION AND CONCAVIFICATION METHODS FOR SOME GLOBAL OPTIMIZATION PROBLEMS

In this paper, firstly, we propose several convexification and concavification transformations to convert a strictly monotone function into a convex or concave function, then we propose several convexification and concavification transformations to convert a non-convex and non-concave objective function into a convex or concave function in the programming problems with convex or concave constraint functions, and propose several convexification and concavification transformations to convert a non-monotone objective function into a convex or concave function in some programming problems with strictly monotone constraint functions. Finally, we prove that the original programming problem can be converted into an equivalent concave minimization problem, or reverse convex programming problem or canonical D.C. programming problem. Then the global optimal solution of the original problem can be obtained by solving the converted concave minimization problem, or reverse convex programming problem or canonical D.C. programming problem using the existing algorithms about them.

On Convexification for a Class of Global Optimization Problems

In this paper,firstly,we give a counterexample to point out there exist deficiencies in our previous works(Wu et al.in J Glob Optim 31:45-60,2005).In addition,we improve the corresponding results.Finally,an example is presented to illustrate how a monotone non-convex optimization problem can be transformed into an equivalent convex minimization problem.

Strong observability as a sufficient condition for non-singularity and lossless convexification in optimal control with mixed constraints

This paper analyzes optimal control problems with linear time-varying dynamics defined on a smooth manifold in addition to mixed constraints and pure control constraints.The main contribution is the identification of sufficient conditions for the optimal controls to be non-singular,which enables exact(or lossless)convex relaxations of the control constraints.The problem is analyzed in a geometric framework using a recent maximum principle on manifolds,and it is shown that strong observability of the dual system on the cotangent space is the key condition.Two minimum time problems are analyzed and solved.A minimum fuel planetary descent problem is then analyzed and relaxed to a convex form.Convexity enables its efficient solution in less than one second without any initial guess.

非光滑多目标优化问题中KT乘子集的非空有界性

运用次微分convexificator提出约束规格并研究具有不等式和集合约束的局部Lipschitz多目标优化问题KT乘子集的非空有界性,得到了在局部弱有效解处所提出的约束规格是KT乘子集非空有界的充分必要条件.

关于次微分convexificator与exhauster关系的一个结果

从次微分convexificator和exhauster的概念出发,在局部Lipschitz条件下,应用函数的上凸(下凹)逼近的exhaustive族存在性定理,并结合已有的相关结论,得出必存在上exhauster和上半正则convexificator,分别记为E*h和坠*f(x),使得坠*f(x)∈E*h.

Some Convexificators-Based Optimality Conditions for Nonsmooth Mathematical Program with Vanishing Constraints

In this paper, by using the notion of convexificator, we introduce the generalized standard Abadie constraint qualification and the generalized MPVC Abadie constraint qualification, and define the generalized stationary conditions for the nonsmooth mathematical program with vanishing constraints (MPVC for short). We show that the generalized strong stationary is the first order necessary optimality condition for nonsmooth MPVC under the generalized standard Abadie constraint qualification. Sufficient conditions for global or local optimality for nonsmooth MPVC are also derived under some generalized convexity assumptions.

Survey of convex optimization for aerospace applications

Convex optimization is a class of mathematical programming problems with polynomial complexity for which state-of-the-art, highly efficient numerical algorithms with predeterminable computational bounds exist. Computational efficiency and tractability in aerospace engineering, especially in guidance, navigation, and control (GN&C), are of paramount importance. With theoretical guarantees on solutions and computational efficiency, convex optimization lends itself as a very appealing tool. Coinciding the strong drive toward autonomous operations of aerospace vehicles, convex optimization has seen rapidly increasing utility in solving aerospace GN&C problems with the potential for onboard real-time applications. This paper attempts to provide an overview on the problems to date in aerospace guidance, path planning, and control where convex optimization has been applied. Various convexification techniques are reviewed that have been used to convexify the originally nonconvex aerospace problems. Discussions on how to ensure the validity of the convexification process are provided. Some related implementation issues will be introduced as well.

Convex optimization of virtual storage system scheduling in market environment

Due to the popularization of distributed energy resources(DERs),the aggregated prosumer effect excels a general energy storage system characteristic.Virtual energy storage system(VESS)concept is proposed hereby that mimics an actual storage unit and incorporates the same charging(consumer)and discharging(producer)modes.It is possible to provide ancillary services via VESS by exploiting the flexibility and thus much research has been proposed on the optimization of the VESS scheduling.In general,the charging and discharging efficiencies of VESS are different and there can be only one status at a time slot.To achieve the optimal schedule while considering the constraints above,binary terms should be introduced into the optimization problem which end up with a nonconvex problem.In this paper,a complimentary mathematical proof is given for the convexification of this mixed-integer linear programming(MILP)problem so that the linear programming(LP)method can be applied instead if the objective function is linear.The proposed proof is validated through a case study and the simulation results show the effectiveness of the proposed method.

A STRONG OPTIMIZATION THEOREM IN LOCALLY CONVEX SPACES

This paper presents a geometric characterization of convex sets in locally convex spaces onwhich a strong optimization theorem of the Stegall-type holds, and gives Collier's theorem ofw* Asplund spaces a localized setting.

Image meshing via hierarchical optimization

Vector graphic, as a kind of geometric representation of raster images, has many advantages, e.g.,definition independence and editing facility. A popular way to convert raster images into vector graphics is image meshing, the aim of which is to find a mesh to represent an image as faithfully as possible. For traditional meshing algorithms, the crux of the problem resides mainly in the high non-linearity and non-smoothness of the ob jective,which makes it difficult to find a desirable optimal solution. To ameliorate this situation, we present a hierarchical optimization algorithm solving the problem from coarser levels to finer ones, providing initialization for each level with its coarser ascent. To further simplify the problem, the original non-convex problem is converted to a linear least squares one, and thus becomes convex, which makes the problem much easier to solve. A dictionary learning framework is used to combine geometry and topology elegantly. Then an alternating scheme is employed to solve both parts. Experiments show that our algorithm runs fast and achieves better results than existing ones for most images.

On Semi-infinite Mathematical Programming Problems with Equilibrium Constraints Using Generalized Convexity

In this paper,we consider semi-infinite mathematical programming problems with equilibrium constraints(SIMPPEC).By using the notion of convexificators,we establish sufficient optimality conditions for the SIMPPEC.We formulate Wolfe and Mond–Weir-type dual models for the SIMPPEC under the invexity and generalized invexity assumptions.Weak and strong duality theorems are established to relate the SIMPPEC and two dual programs in the framework of convexificators.