Visualization is often invaluable to understand the behavior of optimization algorithms, identify their bottlenecks or pathological behaviors, and suggest remedial techniques. Yet developing visualizations is often a tedious activity requiring significant time and expertise. This paper presents a framework for the visualization of constraint-based local search (CBLS) algorithms. Given a high-level model and a declarative visualization specification, the CBLS visualizer systematically produces animations to visualize constraints and objectives, violations, and conflicts, as well as the temporal behavior of these measures. The visualization specification is declarative and typically composed of a triple (what,where,how) indicating what to display, where, and with which graphical objects. The visualizer architecture is compositional and extensible. It provides building blocks which can be assembled freely by the user and focuses almost exclusively on static aspects, the dynamic aspects being automated by the use of invariants. The paper highlights various functionalities of the visualizer and describes a blueprint for its implementation.