Two-Stage Optimization Process

• The two-stage optimization process is defined by initial 'here-and-now' decisions (reservations) followed by recourse or revocation actions once uncertainty is revealed.
• It employs an LP-based approximation framework with double randomized rounding to achieve provable performance bounds in covering and facility location problems.
• The model’s flexibility enhances real-world applications like network design and facility location, where early commitments can be partially adjusted or revoked.

A two-stage optimization process is an approach in mathematical programming where decisions are partitioned into two sequential sets: the first-stage ("here-and-now") decisions are made before the realization of uncertain or scenario-dependent data, and the second-stage ("wait-and-see" or "recourse") decisions are made after the uncertainty is revealed. The classical model is substantially extended by the incorporation of recourse and revocation, as introduced in the two-stage optimization with recourse and revocation framework (Jiang, 2016). This variant enables partial withdrawal or cancellation ("revocation") of first-stage commitments after the scenario is observed, often with a partial refund or adjusted cost, which more accurately represents real-world planning under uncertainty where some flexibility in early commitments is realistic.

1. Model Structure: Recourse and Revocation Extension

The canonical two-stage optimization with recourse is generalized by allowing the decision-maker not only to adjust (recourse) but also to revoke some first-stage decisions ("recourse and revocation" or "recourse and reservation" model). Formally, the process comprises:

• First Stage (Reservation): A set of resources (e.g., edges in a network, facilities in location problems) are reserved at a fraction (denoted $0 < o < 1$) of their full cost.
• Second Stage (Recourse actions): Upon revelation of the scenario, a subset of previously reserved resources can be "activated" by paying the remaining cost $(1-o)$ per item, or "revoked" with a partial cost reduction ("sale back"), and any additional needed resources can be bought via recourse at an inflated factor $X > 1$.

For instance, in the rooted Steiner tree example, the objective is:

$$\min o\cdot w(F_0) + (1-o)\cdot E[w(F_1)] + X\cdot E[w(F_2)]$$

where $F_0$ is the set of reserved edges, $F_1$ the subset purchased in the scenario, and $F_2$ the edges bought as recourse.

This approach generalizes classical two-stage models, as setting $o=0$ (no reservations) or $o=1$ (no recourse) reverts to standard formulations. The framework is particularly relevant for applications such as network design, facility location, and combinatorial cover problems where early commitments can be partially revised.

2. Linear Programming Formulations and Rounding Schemes

The central technical contribution is the development of an LP-based approximation framework tailored for covering-type problems (e.g., set cover, vertex cover) within the recourse and revocation model.

LP Variable Definitions

• $x_s$ — Fraction reserved of set $s$ in the first stage.
• $y_{a,s}$ — Fraction of set $s$ purchased ("activated") in scenario $a$ ($y_{a,s} \leq x_s$ due to revocation constraints).
• $z_{a,s}$ — Fraction of set $s$ purchased via recourse in scenario $a$ (at inflated cost).

LP relaxation:

$$\begin{align*} \min\ & o \sum_s w_s x_s + (1-o) E_a \left[\sum_s w_s y_{a,s}\right] + X E_a \left[\sum_s w_s z_{a,s}\right] \ \text{s.t. }\ & \sum_{s: e \in s} (y_{a,s} + z_{a,s}) \geq 1,\ \forall e, a \ & y_{a,s} \leq x_s, \quad x_s, y_{a,s}, z_{a,s} \geq 0 \end{align*}$$

Rounding Scheme: Double Randomized Rounding and Preprocessing

Due to dependencies between $x$ and $y$ variables, direct rounding risks violating feasibility ($y_{a,s} \leq x_s$ post rounding). To address this:

• Solution Preprocessing: Scale up fractional variables by a factor $k = (X+o-1)/(2-o)$, adjusting support to ensure $x_s \geq 1/2$ where needed.
• Stage 1 Rounding: Independently select each set $s$ with probability $2 x_s$ (amplification to ensure coverage).
• Stage 2 Rounding: For each scenario, among selected sets, further sample with probability proportional to $y_{a,s}/x_s$. This ensures that no more than the reserved amount is purchased, maintaining coupling between stages.

This yields approximation guarantees of $2k \cdot O(\log n)$ for set cover and $4k$ for vertex cover (where $k$—the scaling factor—depends on model parameters $o$, $X$).

Table: Summary of Rounding Results

Problem	Approximation Guarantee	Main Technique
Set Cover	$2k \cdot O(\log n)$	Double randomized/LP
Vertex Cover	$4k$	Two-stage LP rounding

The generality of the scheme allows many deterministic LP rounding approaches to be adapted to the recourse and revocation model.

3. Application to Facility Location: SUFL with Recourse and Revocation

The stochastic uncapacitated facility location (SUFL) problem is extended to allow for first-stage reservations ($o f_i$ per facility) and recourse activations or additional facility openings after demand scenarios.

SUFL LP Relaxation

Variables:

• $y_i$ — Fractional first-stage reservation
• $y_i^{(a)}$ — Activated fraction (discounted second-stage payment) under scenario $a$
• $z_i^{(a)}$ — Fraction opened as recourse (inflated cost) under $a$
• $x_{ij}^{(a)}$ — Client assignments (with triangle inequality for connection costs)

Objective:

$$\min o \sum_i f_i y_i + (1-o) \sum_i f_i y_i^{(a)} + X E_a \left[\sum_i f_i z_i^{(a)} + \sum_{i,j} d_{ij} x_{ij}^{(a)}\right]$$

Constraints ensure every client is assigned and only assigned to opened/activated facilities.

Approximation Algorithmic Approach

• Filtering: For each assignment, identify neighborhoods such that high fractional assignment is contained within small cost balls around clients.
• Clustering: Order clients by service cost and form clusters sharing common candidate facilities, so opening one suffices per cluster, amortizing the opening costs.
• Dependent Rounding: Use a coupling so that each cluster has at least one reserved facility, and non-clustered facilities are treated independently.
• Activation/Recourse: In each scenario, activate reserved facilities probabilistically (depending on LP solution); any remaining demand is covered at recourse cost.

Approximation guarantees:

• 5-approximation in the general case with clustering/filtering (matching the order of known results for basic SUFL).
• Improved to 3.81-approximation for favorable parameter regimes (e.g., large $o$), combining with advanced deterministic algorithms.

4. Theoretical and Algorithmic Implications

The recourse and revocation extension preserves core analytic tractability. Key analytical advances include:

• Generalization: Setting $o=0$ or disabling revocation returns the standard two-stage model; thus, all existing deterministic and two-stage stochastic approaches are special cases.
• Extension of Deterministic Methods: Most LP rounding schemes, clustering, and dependent rounding techniques from combinatorial optimization directly transfer (with preprocessing).
• Performance Bounds: Approximation ratios are of the same order as their deterministic or classical two-stage counterparts, up to model-dependent factors reflecting the partial reservation and recourse/activation cost inflation.

This extension enables the modeling of a broad spectrum of practical scenarios where commitments can be partially withdrawn, not just augmented.

5. Representative Mathematical Formulations

For set cover:

$$\min o \sum_s w_s x_s + (1-o) E_a \left[\sum_s w_s y_{a,s}\right] + X E_a \left[\sum_s w_s z_{a,s}\right],$$

subject to

$$\sum_{s: e \in s} (y_{a,s} + z_{a,s}) \geq 1\ ,\quad y_{a,s} \leq x_s\ ,\quad x_s, y_{a,s}, z_{a,s} \geq 0.$$

For SUFL:

$$\min o \sum_{i\in F} f_i y_i + (1-o)\sum_{i\in F} f_i y_i^{(a)} + X \sum_a p_a \left(\sum_{i\in F} f_i z_i^{(a)} + \sum_{i\in F, j\in C} d_{ij} x_{ij}^{(a)}\right)$$

with scenario-wise assignment and linking constraints.

6. Connections to Related Optimization Frameworks

This two-stage model with revocation connects to:

• Classical two-stage stochastic optimization via a choice of $o=0$;
• Deterministic combinatorial optimization as a zero-uncertainty or one-scenario limit;
• Covering and facility location: Classical LP rounding and clustering methods apply with modifications for linking variables and revocation consistency;
• Scenario-based planning: The polynomial scenario assumption is crucial to achieving computational efficiency.

7. Practical Applications and Algorithmic Implementation

Key practical insights from the model are:

• Efficiency in Approximation: The double randomized rounding and clustering strategies offer not only theoretical approximation guarantees but also efficient implementability (e.g., in $O(\log n)$ or constant-factor time per instance).
• Scenario Management: Polynomial scenario assumption is required for tractable LP rounding schemes. For large scenario sets, further decomposition or scenario reduction may be warranted.
• Applicability to Real-World Problems: Network design, facility location with possible project rollbacks, dynamic resource planning, and any context where reservation decisions may be partially or fully revoked upon observing demand or uncertainty realizations.

The inclusion of revocation more accurately reflects real-world flexibility and yields algorithms with provable guarantees analogous to their classical deterministic and stochastic optimization antecedents.

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In conclusion, the two-stage optimization process with recourse and revocation substantially enriches traditional stochastic combinatorial optimization, offering a framework that simultaneously accommodates advance commitment, reactive adjustment, and partial withdrawal with a provably efficient algorithmic toolkit (Jiang, 2016).