Cutting Plane Methods and Subgradient Methods

Abstract

Interior point methods have proven very successful at solving linear programming problems. When an explicit linear programming formulation is either not available or is too large to employ directly, a column generation approach can be used. Examples of column generation approaches include cutting plane methods for integer programming and decomposition methods for many classes of optimization problems. This tutorial discusses the use of interior point methods in a column generation scheme. Semidefinite programming relaxations of combinatorial optimization problems are often tighter than linear programming relaxations. This tutorial describes some research in using semidefinite programming relaxations to find exact solutions to combinatorial optimization problems. Semidefinite programs are expensive to solve directly, so cutting surface approaches to solving them are also considered. Finally, this tutorial looks at recent smoothing techniques for solving nonsmooth optimization problems using a subgradient approach. These methods have some links to cutting surface approaches.

From the separation to the intersection sub-problem in Benders decomposition models with prohibitively-many constraints

“…As [10] put it, this algorithm has a lack of stability that had been noted for a long time, with intermediate "solutions possibly moving dramatically after adding cutting planes". Experiments suggest that our Benders Cutting-Planes algorithms do suffer from such issues, i.e., the optimal solution at a given iteration might share very few features (selected edges) with the optimal solution at the previous iteration.…”

“…At each iteration, we fist solve the intersection sub-problem on this query point r m . If the returned cut separates the current optimal solution r, we consider the smoothed intersection is successful (a hit) and we no longer apply the intersection sub-problem on r. Otherwise, the smoothed cut is unsuccessful and we need to call a second intersection sub-problem on r. The use of the best feasible solution to define the query point is reminiscent of centralization methods (or centering schemes), in which one uses more interior solutions as query pointse.g., see references on the analytic-center Cutting-Planes method in [10,11].…”

“…The new method has less successful smoothed cuts (Column 13), meaning that it often has to generate two constraints at each iteration, leading to (relaxed) master ILPs with more generated constraints, increasing the CPU time. 10 A natural question that arises is whether the new method achieves similar or better speed-ups on larger graphs. On the first random-10 instances of Table 2 one can simply note the following CPU time speed-ups: 1.33 for the instance id = 1 with |E| = 1000, 2.27 for the instance id = 1 with |E| = 1500 and 3.27 for the instance id = 1 with |E| = 2000.…”

“…When setting |E| ≥ 100, both methods have high chances of failing -notice the last two rows in which we prefer to provide the success rates instead of detailed CPU time statistics. We will 10 Let us exemplify this on the first instance of Table 2. The standard Cutting-Planes requires an average of 271 iterations and the smoothed constraint is sufficient in most of them (next-to-last column indicates an average of 264); the remaining 271-264 iterations need two constraints, leading to a total of 264+2×(271-264)=276 generated constraints along all iterations.…”

Convergence and polynomiality of primal-dual interior-point algorithms for linear programming with selective addition of inequalities

“…Although inequalities can be converted to equalities using nonnegative slack variables in principle, this reformulation is generally inefficient if only few inequalities are active at optimality, and impractical if their number is much larger than the number of original variables. Common examples are semiinfinite optimization problems and continuous relaxations of combinatorial problems, where large numbers of inequalities are typically handled most efficiently using column-generation and cutting-plane methods [12,27,28] or other dual approaches including augmented Lagrangian relaxations [8,15,17]. Starting from an initial relaxation, the conventional scheme of these methods is to repeatedly solve and update successive relaxations by adding new inequalities that are violated at the currently optimal point, until the new optimal solution is also feasible for the original problem.…”