Sensitivity analysis in convex quadratic optimization: invariant support set interval

In sensitivity analysis one wants to know how the problem and the optimal solutions change under the variation of the input data. We consider the case when variation happens in the right-hand side of the constraints and/or in the linear term of the objective function. We are interested to find the range of the parameter variation in Convex Quadratic Optimization (CQO) problems where the support set of a given primal optimal solution remains invariant. This question has been first raised in Linear Optimization (LO) and known as Type II (so-called Support Set Invariancy) sensitivity analysis. We present computable auxiliary problems to identify the range of parameter variation in support set invariancy sensitivity analysis for CQO. It should be mentioned that all the given auxiliary problems are LO problems and can be solved by an interior point method in polynomial time. We also highlight the differences between the characteristics of support set invariancy sensitivity analysis for LO and CQO.     

Sensitivity analysis in linear optimization: Invariant support set intervals

Sensitivity analysis is one of the most interesting and preoccupying areas in optimization. Many attempts are made to investigate the problem’s behavior when the input data changes. Usually variation occurs in the right hand side of the constraints and/or the objective function coefficients. Degeneracy of optimal solutions causes considerable difficulties in sensitivity analysis. In this paper we briefly review three types of sensitivity analysis and consider the question: what is the range of the parameter, where for each parameter value, an optimal solution exists with exactly the same set of positive variables that the current optimal solution has. This problem is coming from managerial requirements. Managers want to know in what range of variation of sources or prices in the market can they keep the installed production lines active, and only production’s levels would change.     

Uniform saturation in linear inequality systems

Redundant constraints in linear inequality systems can be characterized as those inequalities that can be removed from an arbitrary linear optimization problem posed on its solution set without modifying its value and its optimal set. A constraint is saturated in a given linear optimization problem when it is binding at the optimal set. Saturation is a property related with the preservation of the value and the optimal set under the elimination of the given constraint, phenomena which can be seen as weaker forms of excess information in linear optimization problems. We say that an inequality of a given linear inequality system is uniformly saturated when it is saturated for any solvable linear optimization problem posed on its solution set. This paper characterizes the uniform saturated inequalities and other related classes of inequalities. This work was supported by the MCYT of Spain and FEDER of UE, Grant BFM2002-04114-C02-01.     

A Lower Bound for the Penalty Parameter in the Exact Minimax Penalty Function Method for Solving Nondifferentiable Extremum Problems

In the paper, we consider the exact minimax penalty function method used for solving a general nondifferentiable extremum problem with both inequality and equality constraints. We analyze the relationship between an optimal solution in the given constrained extremum problem and a minimizer in its associated penalized optimization problem with the exact minimax penalty function under the assumption of convexity of the functions constituting the considered optimization problem (with the exception of those equality constraint functions for which the associated Lagrange multipliers are negative—these functions should be assumed to be concave). The lower bound of the penalty parameter is given such that, for every value of the penalty parameter above the threshold, the equivalence holds between the set of optimal solutions in the given extremum problem and the set of minimizers in its associated penalized optimization problem with the exact minimax penalty function.     

A Branch-and-Bound Approach for Solving a Class of Generalized Semi-infinite Programming Problems

A nonconvex generalized semi-infinite programming problem is considered, involving parametric max-functions in both the objective and the constraints. For a fixed vector of parameters, the values of these parametric max-functions are given as optimal values of convex quadratic programming problems. Assuming that for each parameter the parametric quadratic problems satisfy the strong duality relation, conditions are described ensuring the uniform boundedness of the optimal sets of the dual problems w.r.t. the parameter. Finally a branch-and-bound approach is suggested transforming the problem of finding an approximate global minimum of the original nonconvex optimization problem into the solution of a finite number of convex problems.     

Value-Estimation Function Method for Constrained Global Optimization

A novel value-estimation function method for global optimization problems with inequality constraints is proposed in this paper. The value-estimation function formulation is an auxiliary unconstrained optimization problem with a univariate parameter that represents an estimated optimal value of the objective function of the original optimization problem. A solution is optimal to the original problem if and only if it is also optimal to the auxiliary unconstrained optimization with the parameter set at the optimal objective value of the original problem, which turns out to be the unique root of a basic value-estimation function. A logarithmic-exponential value-estimation function formulation is further developed to acquire computational tractability and efficiency. The optimal objective value of the original problem as well as the optimal solution are sought iteratively by applying either a generalized Newton method or a bisection method to the logarithmic-exponential value-estimation function formulation. The convergence properties of the solution algorithms guarantee the identification of an approximate optimal solution of the original problem, up to any predetermined degree of accuracy, within a finite number of iterations.     

Effects of change of scale on optimality in a scheduling model with priorities and earliness/tardiness penalties

We consider the effect of changes of scale of measurement on the conclusion that a particular solution to a scheduling problem is optimal. The analysis in this paper was motivated by the problem of finding the optimal transportation schedule when there are penalties for both late and early arrivals, and when different items that need to be transported receive different priorities. We note that in this problem, if attention is paid to how certain parameters are measured, then a change of scale of measurement might lead to the anomalous situation where a schedule is optimal if the parameter is measured in one way, but not if the parameter is measured in a different way that seems equally acceptable. This conclusion about the sensitivity of the conclusion that a given solution to a combinatorial optimization problem is optimal is different from the usual type of conclusion in sensitivity analysis, since it holds even though there is no change in the objective function, the constraints, or other input parameters, but only in scales of measurement. We emphasize the need to consider such changes of scale in analysis of scheduling and other combinatorial optimization problems. We also discuss the mathematical problems that arise in two special cases, where all desired arrival times are the same and the simplest case where they are not, namely the case where there are two distinct arrival times but one of them occurs exactly once. While specialized, these two examples illustrate the types of mathematical problems that arise from considerations of the interplay between scale-types and optimization.     

带消失约束的区间值优化问题的最优性条件与对偶定理

本文考虑一类带消失约束的非光滑区间值优化问题(IOPVC)。在一定的约束条件下得到了问题(IOPVC)的LU最优解的必要和充分性最优性条件,研究了其与Mond-Weir型对偶模型和Wolfe型对偶模型之间的弱对偶,强对偶和严格逆对偶定理,并给出了一些例子来阐述我们的结果。     

Dual Bounds and Optimality Cuts for All-Quadratic Programs with Convex Constraints

A central problem of branch-and-bound methods for global optimization is that often a lower bound do not match with the optimal value of the corresponding subproblem even if the diameter of the partition set shrinks to zero. This can lead to a large number of subdivisions preventing the method from terminating in reasonable time. For the all-quadratic optimization problem with convex constraints we present optimality cuts which cut off a given local minimizer from the feasible set. We propose a branch-and-bound algorithm using optimality cuts which is finite if all global minimizers fulfill a certain second order optimality condition. The optimality cuts are based on the formulation of a dual problem where additional redundant constraints are added. This technique is also used for constructing tight lower bounds. Moreover we present for the box-constrained and the standard quadratic programming problem dual bounds which have under certain conditions a zero duality gap.     

Semicontinuity and convergence for vector optimization problems with approximate equilibrium constraints

This paper mainly intends to present some semicontinuity and convergence results for perturbed vector optimization problems with approximate equilibrium constraints. We establish the lower semicontinuity of the efficient solution mapping for the vector optimization problem with perturbations of both the objective function and the constraint set. The constraint set is the set of approximate weak efficient solutions of the vector equilibrium problem. Moreover, upper Painlevé–Kuratowski convergence results of the weak efficient solution mapping are showed. Finally, some applications to the optimization problems with approximate vector variational inequality constraints and the traffic network equilibrium problems are also given. Our main results are different from the ones in the literature.     

On the global optimality of generalized trust region subproblems

Quadratically constrained quadratic programming is an important class of optimization problems. We consider the case with one quadratic constraint. Since both the objective function and its constraint can be neither convex nor concave, it is also known as the ‘generalized trust region subproblem.’ The theory and algorithms for this problem have been well studied under the Slater condition. In this article, we analyse the duality property between the primal problem and its Lagrangian dual problem, and discuss the attainability of the optimal primal solution without the Slater condition. The relations between the Lagrangian dual and semidefinite programming dual is also given.     

Bilevel problems over polyhedra with extreme point optimal solutions

Bilevel programming involves two optimization problems where the constraint region of the upper level problem is implicitly determined by another optimization problem. In this paper we focus on bilevel problems over polyhedra with upper level constraints involving lower level variables. On the one hand, under the uniqueness of the optimal solution of the lower level problem, we prove that the fact that the objective functions of both levels are quasiconcave characterizes the property of the existence of an extreme point of the polyhedron defined by the whole set of constraints which is an optimal solution of the bilevel problem. An example is used to show that this property is in general violated if the optimal solution of the lower level problem is not unique. On the other hand, if the lower level objective function is not quasiconcave but convex quadratic, assuming the optimistic approach we prove that the optimal solution is attained at an extreme point of an ??enlarged?? polyhedron.     

离散椭球分布下两阶段WCVaR风险利润优化模型及应用

本文研究随机变量非完全分布下的两阶段风险-利润优化问题。采用最坏情况下条件风险(Worst-case Conditional Value-at-Risk:WCVaR) 度量指标,在离散椭球分布下建立了两阶段WCVaR 约束下利润期望最大优化模型,运用优化对偶方法将复杂的Max-Min 结构化简,理论上证明了简化模型和原模型的同解性,以发电商电能分配组合优化为数值实例,验证了模型和计算方法的有效性。     

Quantitative stability analysis of stochastic quasi-variational inequality problems and applications

We consider a parametric stochastic quasi-variational inequality problem (SQVIP for short) where the underlying normal cone is defined over the solution set of a parametric stochastic cone system. We investigate the impact of variation of the probability measure and the parameter on the solution of the SQVIP. By reformulating the SQVIP as a natural equation and treating the orthogonal projection over the solution set of the parametric stochastic cone system as an optimization problem, we effectively convert stability of the SQVIP into that of a one stage stochastic program with stochastic cone constraints. Under some moderate conditions, we derive Hölder outer semicontinuity and continuity of the solution set against the variation of the probability measure and the parameter. The stability results are applied to a mathematical program with stochastic semidefinite constraints and a mathematical program with SQVIP constraints.     

Random algorithms for convex minimization problems

This paper deals with iterative gradient and subgradient methods with random feasibility steps for solving constrained convex minimization problems, where the constraint set is specified as the intersection of possibly infinitely many constraint sets. Each constraint set is assumed to be given as a level set of a convex but not necessarily differentiable function. The proposed algorithms are applicable to the situation where the whole constraint set of the problem is not known in advance, but it is rather learned in time through observations. Also, the algorithms are of interest for constrained optimization problems where the constraints are known but the number of constraints is either large or not finite. We analyze the proposed algorithm for the case when the objective function is differentiable with Lipschitz gradients and the case when the objective function is not necessarily differentiable. The behavior of the algorithm is investigated both for diminishing and non-diminishing stepsize values. The almost sure convergence to an optimal solution is established for diminishing stepsize. For non-diminishing stepsize, the error bounds are established for the expected distances of the weighted averages of the iterates from the constraint set, as well as for the expected sub-optimality of the function values along the weighted averages.     

Optimal discrete-valued control computation

In this paper, we consider an optimal control problem in which the control takes values from a discrete set and the state and control are subject to continuous inequality constraints. By introducing auxiliary controls and applying a time-scaling transformation, we transform this optimal control problem into an equivalent problem subject to additional linear and quadratic constraints. The feasible region defined by these additional constraints is disconnected, and thus standard optimization methods struggle to handle these constraints. We introduce a novel exact penalty function to penalize constraint violations, and then append this penalty function to the objective. This leads to an approximate optimal control problem that can be solved using standard software packages such as MISER. Convergence results show that when the penalty parameter is sufficiently large, any local solution of the approximate problem is also a local solution of the original problem. We conclude the paper with some numerical results for two difficult train control problems.     

A note on strong duality in convex semidefinite optimization: necessary and sufficient conditions

A strong duality which states that the optimal values of the primal convex problem and its Lagrangian dual problem are equal (i.e. zero duality gap) and the dual problem attains its maximum is a corner stone in convex optimization. In particular it plays a major role in the numerical solution as well as the application of convex semidefinite optimization. The strong duality requires a technical condition known as a constraint qualification (CQ). Several CQs which are sufficient for strong duality have been given in the literature. In this note we present new necessary and sufficient CQs for the strong duality in convex semidefinite optimization. These CQs are shown to be sharper forms of the strong conical hull intersection property (CHIP) of the intersecting sets of constraints which has played a critical role in other areas of convex optimization such as constrained approximation and error bounds. Research was partially supported by the Australian Research Council. The author is grateful to the referees for their helpful comments     

Duality and existence theory for nondifferentiable programming

Necessary and sufficient conditions of optimality are given for a nonlinear nondifferentiable program, where the constraints are defined via closed convex cones and their polars. These results are then used to obtain an existence theorem for the corresponding stationary point problem, under some convexity and regularity conditions on the functions involved, which also guarantee an optimal solution to the programming problem. Furthermore, a dual problem is defined, and a strong duality theorem is obtained under the assumption that the constraint sets of the primal and dual problems are nonempty.     

Characteristics of semi-convex frontier optimization

We study semi-convex frontier (SCF) optimization problems where objective functions can be semi-convex and constraint sets can be non-polyhedron, which stem from a growing range of optimization applications such as frontier analysis, multi-objective programming in economics. The new findings of this paper can be summarized as follows: (1) We characterize non-dominated points of a non-polyhedron optimal solution set of a semi-convex frontier program. (2) We obtain optimality conditions of a constant modulus SCF program, of which the objective function is semi-convex with a constant semiconvexity modulus. (3) We obtain a non-smooth Hölder stability of the optimal solutions of a semiconvex frontier program. (4) We use generalized differentiability to establish sensitivity analysis of the optimal value function of a semi-convex frontier program.