Reverse Convex Constraint Research Articles

On tackling reverse convex constraints for non-overlapping of unequal circles

We study the unequal circle-circle non-overlapping constraints, a form of reverse convex constraints that often arise in optimization models for cutting and packing applications. The feasible region induced by the intersection of circle-circle non-overlapping constraints is highly non-convex, and standard approaches to construct convex relaxations for spatial branch-and-bound global optimization of such models typically yield unsatisfactory loose relaxations. Consequently, solving such non-convex models to guaranteed optimality remains extremely challenging even for the state-of-the-art codes. In this paper, we apply a purpose-built branching scheme on non-overlapping constraints and utilize strengthened intersection cuts and various feasibility-based tightening techniques to further tighten the model relaxation. We embed these techniques into a branch-and-bound code and test them on two variants of circle packing problems. Our computational studies on a suite of 75 benchmark instances yielded, for the first time in the open literature, a total of 54 provably optimal solutions, and it was demonstrated to be competitive over the use of the state-of-the-art general-purpose global optimization solvers.

The complexity results of the sparse optimization problems and reverse convex optimization problems

In this paper, we study complexity results of sparse optimization problems and reverse convex optimization problems. These problems are very important subjects of optimization problems. We prove that the complexity result of the sparsity constraint problem and sparse solution problem are all NP-hard in the strong sense and even testing feasibility of the sparsity constraint is NP-complete in the strong sense. Then the sparse optimization problem is NP-hard in the strong sense. We also prove that the reverse convex problem is NP-hard in the strong sense by transforming the sparsity constraint into a reverse convex constraint.

Solving nonconvex nonlinear programs with reverse convex constraints by sequential linear programming

AbstractThe sequential linear programming (SLP) method for solving nonlinear problems was introduced in the 1960s. Many papers that attempted to use SLP reported poor performance and convergence issues. We found that nonlinear programs with reverse convex constraints, which are the most difficult nonlinear programs with many local optima, are solved (heuristically) very well by SLP. We proved that for this type of problems, the solutions to the sequence of the linear programming problems converge to a local optimum. Since the final solution depends on the starting solution, we propose to apply SLP in a multistart approach starting from randomly generated solutions. This multistart SLP is very easy to implement. We recommend that the research community reconsiders the application of SLP for this type of problems.

Mixed-integer bilevel representability

We study the representability of sets by extended formulations using mixed-integer bilevel programs. We show that feasible regions modeled by continuous bilevel constraints (with no integer variables), complementarity constraints, and polyhedral reverse convex constraints are all finite unions of polyhedra. Conversely, any finite union of polyhedra can be represented using any one of these three paradigms. We then prove that the feasible region of bilevel problems with integer variables exclusively in the upper level is a finite union of sets representable by mixed-integer programs and vice versa. Further, we prove that, up to topological closures, we do not get additional modeling power by allowing integer variables in the lower level as well. To establish the last statement, we prove that the family of sets that are finite unions of mixed-integer representable sets (up to topological closures) forms an algebra of sets; i.e., this family is closed under finite unions, intersections and complementation.

Solving Reverse Convex Programming Problems Using a Generalized Cutting-Plane Method

It is well known that each convex function $$f:{\mathbb{R}}^n \longrightarrow {\mathbb{R}}$$ is supremally generated by affine functions. More precisely, each convex function $$f:{\mathbb{R}}^n \longrightarrow {\mathbb{R}}$$ is the upper envelope of its affine minorants. In this paper, we propose an algorithm for solving reverse convex programming problems by using such a representation together with a generalized cutting-plane method. Indeed, by applying this representation, we solve a sequence of problems with a smaller feasible set, in which the reverse convex constraint is replaced by a still reverse convex but polyhedral constraint. Moreover, we prove that the proposed algorithm converges, under suitable assumptions, to an optimal solution of the original problem. This algorithm is coded in MATLAB language and is evaluated by some numerical examples.

On the tuning of predictive controllers: Application of generalized Benders decomposition to the ELOC problem

This work investigates the computational procedures used to obtain global solution to the economic linear optimal control (ELOC) problem. The proposed method employs the generalized Benders decomposition (GBD) algorithm. Compared to the previous branch and bound approach, a naive application of GBD to the ELOC problem will improve computational performance, due to less frequent calls to computationally slow semi-definite programming (SDP) routines. However, the reverse-convex constraints of the original problem will reappear in the relaxed master problem. In response, a convexification of the relaxed master constraints has been developed and proven to preserve global solution characteristics. The result is a multi-fold improvement in computational performance. A technological benefit of decomposing the problem into steady-state and dynamic parts is the ability to utilize nonlinear steady-state models, since the relaxed master problem is free of SDP type constraints and can be solved using any global nonlinear programming algorithm.

Optimality conditions for DC programming problems with reverse convex constraints

In this paper, we give necessary and sufficient optimality conditions for DC programming problems with reverse convex constraints defined by a finite family of pseudoconvex functions. As application, we obtain necessary and sufficient optimality conditions for DC programming problems and fractional programming problems.

Nonsmooth Optimization for Efficient Beamforming in Cognitive Radio Multicast Transmission

It is known that the design of optimal transmit beamforming vectors for cognitive radio multicast transmission can be formulated as indefinite quadratic optimization programs. Given the challenges of such nonconvex problems, the conventional approach in literature is to recast them as convex semidefinite programs (SDPs) together with rank-one constraints. Then, these nonconvex and discontinuous constraints are dropped allowing for the realization of a pool of relaxed candidate solutions, from which various randomization techniques are utilized with the hope to recover the optimal solutions. However, it has been shown that such approach fails to deliver satisfactory outcomes in many practical settings, wherein the determined solutions are found to be unacceptably far from the actual optimality. On the contrary, we in this contribution tackle the aforementioned optimal beamforming problems differently by representing them as SDPs with additional reverse convex (but continuous) constraints. Nonsmooth optimization algorithms are then proposed to locate the optimal solutions of such design problems in an efficient manner. Our thorough numerical examples verify that the proposed algorithms offer almost global optimality whilst requiring relatively low computational load.

D.C.計画問題に対する逆凸制約の緩和について

D.C.計画問題に対する逆凸制約の緩和について

A nonisolated optimal solution for special reverse convex programming problems

In this paper, an efficient algorithm is proposed for globally solving special reverse convex programming problems with more than one reverse convex constraints. The proposed algorithm provides a nonisolated global optimal solution which is also stable under small perturbations of the constraints, and it turns out that such an optimal solution is adequately guaranteed to be feasible and to be close to the actual optimal solution. Convergence of the algorithm is shown and the numerical experiment is given to illustrate the feasibility of the presented algorithm.

Global optimization of process conditions in batch thermal sterilization of food

Conventionally, food is significantly over-processed to ensure safety. Dynamic optimization can be used to compute optimal thermal operation condition to ensure maximum product quality while assuring food safety. Local optimization (LO) algorithms have been used to compute optimal profiles. However, LO is not guaranteed to find the best solution for non-convex functions. We show that the problem can be formulated as a convex problem with a reverse convex constraint and we implement Tuy’s algorithm to optimize globally. The method is deterministic and guaranteed to find the global optimum and therefore it is suitable to evaluate the effectiveness of local optimization to compute global optima. We compared the results of LO and global optimization (GO) to find that GO gives significantly better results for two and three heating time periods. However, for four periods the local optimizer catches up. This suggests that LO is good enough for this problem if we consider strategies with more than four periods implementable. However for many commercial processes less than four heating–cooling stages are used.

On Optimization over the Efficient Set in Linear Multicriteria Programming

The efficient set of a linear multicriteria programming problem can be represented by a reverse convex constraint of the form g(z)≤0, where g is a concave function. Consequently, the problem of optimizing some real function over the efficient set belongs to an important problem class of global optimization called reverse convex programming. Since the concave function used in the literature is only defined on some set containing the feasible set of the underlying multicriteria programming problem, most global optimization techniques for handling this kind of reverse convex constraint cannot be applied. The main purpose of our article is to present a method for overcoming this disadvantage. We construct a concave function which is finitely defined on the whole space and can be considered as an extension of the existing function. Different forms of the linear multicriteria programming problem are discussed, including the minimum maximal flow problem as an example.

Constraint Qualifications and KKT Conditions for Bilevel Programming Problems

In this paper we consider the bilevel programming problem (BLPP), which is a sequence of two optimization problems where the constraint region of the upper-level problem is determined implicitly by the solution set to the lower-level problem. We extend well-known constraint qualifications for nonlinear programming problems such as the Abadie constraint qualification, the Kuhn-Tucker constraint qualification, the Zangwill constraint qualification, the Arrow-Hurwicz-Uzawa constraint qualification, and the weak reverse convex constraint qualification to BLPPs and derive a Karash-Kuhn-Tucker (KKT)-type necessary optimality condition under these constraint qualifications without assuming the lower-level problem satisfying the Mangasarian Fromovitz constraint qualification. Relationships among various constraint qualifications are also given.

Conditions for boundedness in concave programming under reverse convex and convex constraints

In this paper, we are concerned with the problem of boundedness in the constrained global maximization of a convex function. In particular, we present necessary and sufficient conditions for boundedness of a feasible region defined by reverse convex constraints and we establish sufficient and necessary conditions for existence of an upper bound for a convex objective function defined over the system of concave inequality constraints. We also address the problem of boundedness in the global maximization problem when a feasible region is convex and unbounded.

Reverse convex problems: an approach based on optimality conditions

We present some results concerning reverse convex problems. Global optimality conditions for the problems with a nonsmooth reverse convex constraint are established and convergence of an algorithm in the case of linear program with an additional quadratic reverse convex constraint is studied.

Optimality Conditions for D.C. Vector Optimization Problems Under Reverse Convex Constraints

In this paper, we establish global necessary and sufficient optimality conditions for D.C. vector optimization problems under reverse convex constraints. An application to vector fractional mathematical programming is also given.

A deterministic global optimization algorithm for generalized geometric programming

A deterministic global optimization algorithm is proposed for generalized geometric programming (GGP). By utilizing some transformations, the initial non-convex problem is reduced to a reverse convex programming (RCP), where the objective function and constraint functions are convex. Then a linear relaxation of the problem (RCP) is obtained based on the linear lower bounding functions of the convex constraint functions and the linear upper bounding functions of the reverse convex constraint functions inside some hyperrectangle region. A cutting-plane method is proposed to add some effective linear constraints to the linear relaxation programming based on the famous arithmetic-geometric mean inequality, then derive a tighter linear relaxation programming. The proposed global optimization algorithm which connects the branch and bound method with the cutting-plane method successfully is convergent to the global minimum through the successive refinement of the linear relaxation of the feasible region of the objective function and the solutions of a series of linear relaxation problems. And finally the numerical experiment is given to illustrate the feasibility and the robust stability of the present algorithm.

Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints

In this paper we consider a mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with complementarity constraints. Various stationary conditions for MPECs exist in literature due to different reformulations. We give a simple proof to the M-stationary condition and show that it is sufficient for global or local optimality under some MPEC generalized convexity assumptions. Moreover, we propose new constraint qualifications for M-stationary conditions to hold. These new constraint qualifications include piecewise MFCQ, piecewise Slater condition, MPEC weak reverse convex constraint qualification, MPEC Arrow–Hurwicz–Uzawa constraint qualification, MPEC Zangwill constraint qualification, MPEC Kuhn–Tucker constraint qualification, and MPEC Abadie constraint qualification.

Characterizing Set Containments Involving Infinite Convex Constraints and Reverse-Convex Constraints

Dual characterizations of the containment of a closed convex set, defined by infinite convex constraints, in an arbitrary polyhedral set, in a reverse-convex set, defined by convex constraints, and in another convex set, defined by finite convex constraints, are given. A special case of these dual characterizations has played a key role in generating knowledge-based support vector machine classifiers which are powerful tools in data classification and mining. The conditions in these dual characterizations reduce to simple nonasymptotic conditions under Slater's constraint qualification.

A Cutting Plane Algorithm for Linear Reverse Convex Programs

In this paper, global optimization of linear programs with an additional reverse convex constraint is considered. This type of problem arises in many applications such as engineering design, communications networks, and many management decision support systems with budget constraints and economies-of-scale. The main difficulty with this type of problem is the presence of the complicated reverse convex constraint, which destroys the convexity and possibly the connectivity of the feasible region, putting the problem in a class of difficult and mathematically intractable problems. We present a cutting plane method within the scope of a branch-and-bound scheme that efficiently partitions the polytope associated with the linear constraints and systematically fathoms these portions through the use of the bounds. An upper bound and a lower bound for the optimal value is found and improved at each iteration. The algorithm terminates when all the generated subdivisions have been fathomed.